If one sets out to admit only in-place permutations (i.e. the range of permutating characters in string s1 is the same range as in string s2) to lower the 2 t by 1 (if there are more ranges of that type in the arguments given to detect[], 1 is subtracted only once), then the previous function detect[] did not do that. Instead one should use this one:

detect[l1_List, l2_List, o1_Integer, o2_Integer] := 
 Block[{m = Length[l1], permQ = False, x = 0, o3, o41, o42},
  If[o1 < o2,
   (* Contain l1,
   l2 a permutation of more than 2 characters in place? *)
   For[o3 = 1, o3 <= m, o3++,
    If[(o1 <= l1[[o3]] <= o2) && (o1 <= l2[[o3]] <= o2),
     If[x == 0,
      o41 = o3;
      o42 = o3;
      x = 1, (* else *)
      If[o3 == o42 + 1,
       o42 = o3;
       x += 1, (* else *)
       x = 0
       ]
      ], (* else *)
     If[x > 2 && Complement[l1[[o41 ;; o42]], l2[[o41 ;; o42]]] == {},
      permQ = True;
      Break[], (* else *)
      x = 0
      ]
     ]
    ];
   If[permQ,(* "jaro" permutation exists *)
    o2 - o1,(* else *)
    If[x > 2 && Complement[l1[[o41 ;; o42]], l2[[o41 ;; o42]]] == {},
     o2 - o1, (* else *)
     o2 - o1 + 1
     ]
    ], (* else *)
   1
   ]
  ]

with it the test cases are now:

Case 11 has no in-place permutation, therefore it does not lower 2 t and has 2 t one higher than textdistance.jaro. On the other hand, case 8 has an in-place permutation, clearly seen from this illustration

and again - but into the other direction - therefore case 8 lowers 2 t by 1 and has 2 t one lower than textdistance.jaro. As said before, that in-place permutation regulation is needed to bring the "Miss Australia" vs. "Miss Brasilia" example to the same value textdistance.jaro has.

Attachments:

wc-ledesmaJaroDistance.nb

For the sake of completeness, I found a NIST description of the Jaro distance containing

William E. Winkler and Yves Thibaudeau, An Application of the Fellegi-Sunter Model of Record Linkage to the 1990 U.S. Decennial Census, Statistical Research Report Series RR91/09, U.S. Bureau of the Census, Washington, D.C., 1991. The abstract (HTML) and full paper (PDF).

and there it is said on p. 12

Two characters are considered in common only if they are no further apart than (m/2 - 1) where m = max(d,r). Characters in common from two strings are assigned; remaining characters unassigned. Each string has the same number of assigned characters.

That tells you to treat l3 as a whole and suggests the following implementation

Needs["Experimental`"]

Clear[check]
check[l1_List, l2_List] := Flatten[Outer[List, l1, Select[l2, #[[1]] == l1[[1, 1]] &], 1], 1]

Clear[jaro]
jaro[s1_String, s2_String, prec_: $MachinePrecision] := 
 Block[{r, l1 = StringLength[s1], l2 = StringLength[s2], l3, m, t},
   r = Floor[Max[l1, l2]/2] - 1;
   If[r >= 0,(* then *)
    l3 = DeleteDuplicates[
      Flatten[check[{Transpose[{ToCharacterCode[s1], 
              Range[l1]}][[#]]}, 
          Transpose[{ToCharacterCode[s2], Range[l2]}][[
           Min[l2, Max[1, # - r]] ;; Min[l2, # + r]]]] & /@ Range[l1],
        1], ((First[#1] == First[#2]) || (Last[#1] == Last[#2])) &];
    m = Length[l3];
    If[m > 0,
     t = (m - 
         Count[{1, -1} . 
           MapAt[First, 
            SortBy[#, Last] & /@ Transpose[l3], {{1, All}, {2, All}}],
           0])/2;
     N[(m/l1 + m/l2 + (m - t)/m)/3, prec], (* else *)
     0 
     ], (* else *)
    0
    ]
   ] /; StringLength[s1] > 0 && StringLength[s2] > 0 && prec > 1

which has still t sometimes off by 1/2 with respect to the textdistance.jaro. The JaroDistance[] is spurious (this was made with Mathematica 10.3). Taking the drudgery to create a table

In[204]:=  Grid[{{Item["s1"], Item["s2"], Item["td.jaro"], Item["jaro"], 
       Item["Defect"], Item["JaroDistance"]},
      {Item["Miss Australia"], Item["Miss Brasilia"],
       Item[0.8166833166833167],
       Item[jaro["Miss Australia", "Miss Brasilia"], Background -> Pink],
       Item["t+1/2"],
       Item[JaroDistance["Miss Australia", "Miss Brasilia", 
         IgnoreCase -> False], Background -> Pink]
       },
      {Item["DWAYNE"], Item["DUANE"],
       Item[0.8222222222222223],
       Item[jaro["DWAYNE", "DUANE"]],
       Item["-"],
       Item[JaroDistance["DWAYNE", "DUANE", IgnoreCase -> False], 
        Background -> Pink]
       },
      {Item["MARTHA"], Item["MARHTA"],
       Item[0.9444444444444445],
       Item[jaro["MARTHA", "MARHTA"]],
       Item["-"],
       Item[JaroDistance["MARTHA", "MARHTA", IgnoreCase -> False], 
        Background -> Pink]
       },
      {Item["DIXON"], Item["DICKSONX"],
       Item[0.7666666666666666],
       Item[jaro["DIXON", "DICKSONX"]],
       Item["-"],
       Item[JaroDistance["DIXON", "DICKSONX", IgnoreCase -> False], 
        Background -> Pink]
       },
      {Item["JELLYFISH"], Item["SMELLYFISH"],
       Item[0.8962962962962964],
       Item[jaro["JELLYFISH", "SMELLYFISH"]],
       Item["-"],
       Item[JaroDistance["JELLYFISH", "SMELLYFISH", IgnoreCase -> False], 
        Background -> Pink]
       },
      {Item["Miss Mexiko"], Item["Miss Belize"],
       Item[0.7575757575757575],
       Item[jaro["Miss Mexiko", "Miss Belize"]],
       Item["-"],
       Item[JaroDistance["Miss Mexiko", "Miss Belize", 
         IgnoreCase -> False], Background -> Pink]
       },
      {Item["0100010100101001001001001010010"], 
       Item["10000100100111101010101010101010"],
       Item[0.8308371735791091],
       Item[jaro["0100010100101001001001001010010", 
         "10000100100111101010101010101010"]],
       Item["-"],
       Item[JaroDistance["0100010100101001001001001010010", 
         "10000100100111101010101010101010", IgnoreCase -> False]]
       },
      {Item["aasdjkdashdahsgdashdgasj"], Item["asdjkdashdahsgdashdgasj"],
       Item[0.841183574879227],
       Item[jaro["aasdjkdashdahsgdashdgasj", "asdjkdashdahsgdashdgasj"]],
       Item["-"],
       Item[JaroDistance["aasdjkdashdahsgdashdgasj", 
         "asdjkdashdahsgdashdgasj", IgnoreCase -> False], 
        Background -> Pink]
       },
      {Item["abdegopq"], Item["cfhijklmnrstuvwxyz"],
       Item[0.0],
       Item[jaro["abdegopq", "cfhijklmnrstuvwxyz"]],
       Item["-"],
       Item[JaroDistance["abdegopq", "cfhijklmnrstuvwxyz", 
         IgnoreCase -> False]]
       },
      {Item["Mary has a little lamb"], 
       Item["and Meghan has the redhead Harry"],
       Item[0.5631555944055945],
       Item[jaro["Mary has a little lamb", 
         "and Meghan has the redhead Harry"], Background -> Pink],
       Item["t+1/2"],
       Item[JaroDistance["Mary has a little lamb", 
         "and Meghan has the redhead Harry"]]
       },
      {Item["Take[list,-n] gives the last n elements of list."], 
       Item["Take[list,{m,n}] gives elements m through n of list."],
       Item[0.791056166056166],
       Item[jaro["Take[list,-n] gives the last n elements of list.", 
         "Take[list,{m,n}] gives elements m through n of list."], 
        Background -> Pink],
       Item["t+1/2"],
       Item[JaroDistance[
         "Take[list,-n] gives the last n elements of list.", 
         "Take[list,{m,n}] gives elements m through n of list."]]
       }
      }, Background -> {None, {Cyan}}, Frame -> All]

one gets

and I've to confess in complete humbleness that I'm unable to get jaro[] right using the given descriptions what it is meant to be. Please read the cited PDF and tell me.

[1]:William E. Winkler and Yves Thibaudeau, An Application of the Fellegi-Sunter Model of Record Linkage to the 1990 U.S. Decennial

jD = Experimental`JaroDistance;

This function seems not to have a single point of definition:

In[1]:= $Version
Out[1]= "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)"

In[2]:= Needs["Experimental`"]

In[3]:= ?JaroDistance
Experimental`JaroDistance
Attributes[JaroDistance]={Protected}
Options[JaroDistance]={IgnoreCase->False}

In[4]:= JaroDistance["Miss Australia", "Miss Brasilia"]
Out[4]= 0.816683

In[5]:= JaroDistance["Miss Australia", "Miss Brasilia"]
Out[5]= 0.78638

In[6]:= JaroDistance["Miss Australia", "Miss Brasilia", IgnoreCase -> True]
Out[6]= 0.816683

In[7]:= JaroDistance["Miss Australia", "Miss Brasilia", IgnoreCase -> False]
Out[7]= 0.816683

this is mind-boggling; JaroDistance[] has different results under identical input, it has also different result if default option is given or not.

For my part I have made the following Code

which reminds me to the quotation

you witness $FORTRAN66$ in Mathematica. If it is necessary to hammer out the fact that a double loop must be done -- Outer[] states exactly that -- it can be done in Mathematica without using the built-in Symbol For[] - I guess. Let's see whether the Delayed constructs allow for it. Outer[] is an overkill and elegance is of no good if it compromises performance (and/or correctness)

The EditDistance[] is back and

Clear[check, jaro]
check[l1_List, l2_List] := 
 Flatten[Outer[List, l1, Select[l2, #[[1]] == l1[[1, 1]] &], 1], 1]
jaro[s1_String, s2_String, prec_: $MachinePrecision] := 
 Block[{r, l1 = StringLength[s1], l2 = StringLength[s2], l3, m, l4, 
    l5, t},
   r = Floor[Max[l1, l2]/2] - 1;
   If[r >= 0,(* then *)
    l3 = Flatten[
      check[{Transpose[{ToCharacterCode[s1], Range[l1]}][[#]]}, 
         Transpose[{ToCharacterCode[s2], Range[l2]}][[
          Min[l2, Max[1, # - r]] ;; Min[l2, # + r]]]] & /@ Range[l1], 
      1];
    If[Length[l3] > 0,
     {l4, l5} = 
      MapAt[First, 
       SortBy[#, Last] & /@ (Union /@ Transpose[l3]), {{1, All}, {2, 
         All}}];
     m = Min[Length[l4], Length[l5]];
     t = EditDistance[Take[l4, m], Take[l5, m]]/2;
     N[(m/l1 + m/l2 + (m - t)/m)/3, prec], (* else *)
     0 
     ], (* else *)
    0
    ]
   ] /; StringLength[s1] > 0 && StringLength[s2] > 0 && prec > 1

many results of textdistance.jaro() are matched

>>> textdistance.jaro('Miss Australia', 'Miss Brasilia')
0.8166833166833167
>>> textdistance.jaro('Miss Mexiko', 'Miss Belize')
0.7575757575757575
>>> textdistance.jaro('Miss Belize', 'Miss Mexiko')
0.7575757575757575

In[161]:= jaro["Miss Australia", "Miss Brasilia"]
Out[161]= 0.8166833166833167

In[162]:= jaro["Miss Mexiko", "Miss Belize"]
Out[162]= 0.7575757575757576

In[163]:= jaro[ "Miss Belize", "Miss Mexiko"]
Out[163]= 0.7575757575757576

but some not, to be discussed in the following: First define a function jo[] usable in FindRoot[] to find the m and t which have been ssemingly found by textdistance.jaro():

Clear[jo, joex]
jo[s1_Integer, s2_Integer, m_?NumberQ, t_?NumberQ, 
  prec_: $MachinePrecision] := 
 N[(m/s1 + m/s2 + (m - t)/m)/3, prec] /; m > 0 && prec > 0
joex[s1_Integer, s2_Integer, m_Integer, 
  t_?NumberQ] := (m/s1 + m/s2 + (m - t)/m)/3 /; m > 0

so one uses a jaro[] printing it's findings:

>>> textdistance.jaro('Miss Mexikoooooooooooooooooooo', 'Miss Belize')
0.6232323232323232

In[136]:= jaro["Miss Mexikoooooooooooooooooooo", "Miss Belize"]
During evaluation of In[136]:= l4 = Miss Mei
During evaluation of In[136]:= l5 = Miss eie
During evaluation of In[136]:= s1 = 30| s2 = 11| r = 14| m = 8| t = 1
Out[136]= 0.6229797979797980

the result ot textdistance.jaro() follows with m = 7 and t = 0:

In[145]:= jo[30, 11, 7, 0]
Out[145]= 0.6232323232323232

it's unclear how a disagreement in the number of matching characters can happen.

This is interesting, only the first character differs:

>>> textdistance.jaro('aasdjkdashdahsgdashdgasj', 'asdjkdashdahsgdashdgasj')
0.841183574879227

In[146]:= jaro["aasdjkdashdahsgdashdgasj", "asdjkdashdahsgdashdgasj"]
During evaluation of In[146]:= l4 = aasdjkdashdahsgdashdgasj
During evaluation of In[146]:= l5 = asdjkdashdahsgdashdgasj
During evaluation of In[146]:= s1 = 24| s2 = 23| r = 11| m = 23| t = 1
Out[146]= 0.971618357487923

this is reached by the same m (luckily), but t = 10:

In[149]:= jo[24, 23, 23, 10]
Out[149]= 0.8411835748792271

this one has a great discrepance in the number of matching charachters

>>> textdistance.jaro('0100010100101001001001001010010', '10000100100111101010101010101010')
0.8308371735791091

In[67]:= jaro["0100010100101001001001001010010", \
"10000100100111101010101010101010"]
During evaluation of In[67]:= l4 = 0100010100101001001001001010010
During evaluation of In[67]:= l5 = 10000100100111101010101010101010
During evaluation of In[67]:= s1 = 31| s2 = 32| r = 15| m = 31| t = 5
Out[67]= 0.935819892473118

because it's reached with m = 28 and t = 8

In[133]:= jo[31, 32, 28, 16/2]
Out[133]= 0.8308371735791091 

and last but not least, Mary and Meghan

>>> textdistance.jaro('Mary has a little lamb', 'and Meghan has the redhead Harry')
0.5631555944055945

In[167]:= jaro["Mary has a little lamb", "and Meghan has the redhead Harry"]
During evaluation of In[167]:= l4 = Ma has a tte a
During evaluation of In[167]:= l5 = a Meha has the eea a
During evaluation of In[167]:= s1 = 22| s2 = 32| r = 15| m = 14| t = 9/2
Out[167]= 0.5841450216450216

again textdistance,jaro() disagrees with the m (needs 13) as well as with the t (has 4):

In[153]:= jo[22, 32, 13, 4]
Out[153]= 0.5631555944055944

I quit. I expected that the m fits and the difference lies only in determining the t which has an English description

The number of matching (but different sequence order) characters divided by 2 defines the number of transpositions.

as well as a French description

Le nombre de transpositions est obtenu en comparant le i-ème caractère correspondant de {\displaystyle s{1}} s{1} avec le i-ème caractère correspondant de {\displaystyle s{2}} s{2}. Le nombre de fois où ces caractères sont différents, divisé par deux, donne le nombre de transpositions.

but if one does that, even 'Miss Australia' and 'Miss Brasilia' do not agree with textdistance.jaro().

Okay, fix it into

Clear[check, jaro]
check[d_Integer, l1_List, l2_List] := 
  If[l1[[1]] == l2[[1]] && Abs[l1[[2]] - l2[[2]]] <= d,
   {l1, l2}, (* else *) 
   Missing[]
   ];
jaro[s1_String, s2_String] := 
 Block[{r, m = {}, l1 = StringLength[s1], l2 = StringLength[s2], w1, 
    w2, l3, t = 0, l4, l5},
   r = Floor[Max[l1, l2]/2] - 1;
   If[r < 0,
    m = 0, (* else *)
    w1 = Transpose[{ToCharacterCode[s1], Range[l1]}];
    w2 = Transpose[{ToCharacterCode[s2], Range[l2]}];
    l3 = DeleteMissing[
      Flatten[Outer[check[r, #1, #2] &, w1, w2, 1], 1]];
    m = Length[GatherBy[SortBy[l3, First[First[#]] &], First]];
    {l4, l5} = SortBy[#, Last] & /@ (Union /@ Transpose[l3]);
    t = EditDistance[First /@ l4, First /@ l5]/2
    ];
   Print["r = ", r, "| m = ", m, "| t = ", t];
   If[m == 0,
    0, (* else *)
    (m/l1 + m/l2 + (m - t)/m)/3.
    ]
   ] /; StringLength[s1] > 0 && StringLength[s2] > 0

this implementation is still toy because Outer[] goes too far and then ,,,,, using the Levenshtein distance in computing the Jaro distance seems a bit lunatic ... but at least the test cases go through:

In[71]:= jaro["s", "s"]
During evaluation of In[71]:= r = -1| m = 0| t = 0
Out[71]= 0

In[72]:= jaro["Miss", "Miss"]
During evaluation of In[72]:= r = 1| m = 4| t = 0
Out[72]= 1.

In[73]:= jaro @@@ {{"DWAYNE", "DUANE"},
  {"MARTHA", "MARHTA"},
  {"DIXON", "DICKSONX"},
  {"JELLYFISH", "SMELLYFISH"}}
During evaluation of In[73]:= r = 2| m = 4| t = 0
During evaluation of In[73]:= r = 2| m = 6| t = 1
During evaluation of In[73]:= r = 3| m = 4| t = 0
During evaluation of In[73]:= r = 4| m = 8| t = 0
Out[73]= {0.822222, 0.944444, 0.766667, 0.896296}

In[76]:= jaro["Miss Argentina", "Miss Brasilia"]
During evaluation of In[76]:= r = 6| m = 8| t = 3/2
Out[76]= 0.666438

In[77]:= jaro["0100010100101001001001001010010", \
"10000100100111101010101010101010"]
During evaluation of In[77]:= r = 15| m = 31| t = 9/2
Out[77]= 0.941196