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

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