Okkes,
I checked your assertion that they are the same with c1 and c2 and p1 and p2 interchanged by swapping the variables. the minimum is the same but the expression appears a bit different at first glance.
In[21]:= mod2 =
model2 /. p1 -> p3 /. p2 -> p1 /. p3 -> p2 /. c1 -> c3 /. c2 -> c1 /.
c3 -> c2;
In[29]:= fit2 =
NMinimize[{model1, c1 > 0 && c2 > 0 && is1 > 0 && is2 > 0},
parameters] // Chop
Out[29]= {0.0795293, {\[Gamma] -> 12.3339, \[Epsilon] ->
2.01739, \[CapitalDelta]G0\[Sigma] -> 1.54585, c1 -> 21.4073,
c2 -> 4.34306, p1 -> 22.8236, p2 -> 41.89, is1 -> 8.05893,
is2 -> 7.17248}}
In[22]:= fit2 =
NMinimize[{mod2, c1 > 0 && c2 > 0 && is1 > 0 && is2 > 0},
parameters] // Chop
Out[22]= {0.0795293, {\[Gamma] -> 12.904, \[Epsilon] ->
2.04318, \[CapitalDelta]G0\[Sigma] -> 1.61479, c1 -> 27.1908,
c2 -> 4.53967, p1 -> 22.8236, p2 -> 41.8901, is1 -> 7.16493,
is2 -> 7.97499}}
It appears, however, that because the system is nonlinear you are falling into a local minimum. You will need to guide the result by giving starting ranges for one or more of the parameters or add more constraints:
In[28]:= fit2 =
NMinimize[{model2, c1 > 0 && c2 > 0 && is1 > 0 && is2 > 0},
parameters /. c2 -> {c2, 1, 10}] // Chop
Out[28]= {0.0795293, {\[Gamma] -> 19.644, \[Epsilon] ->
2.01061, \[CapitalDelta]G0\[Sigma] -> 2.08409, c1 -> 3.96859,
c2 -> 18.748, p1 -> 41.8901, p2 -> 22.8236, is1 -> 7.17438,
is2 -> 8.07787}}
Note that each solution that gives the same minimum (model1, mod2, and model2 with a starting range on c2) has different combinations of the parameters. This suggests that there are multiple combinations of parameters that have the same minimum. This appears to be an ill conditioned fit. You may need to go back to the physics for some guidance (or post more details about the origin of the expressions).
