The two solutions are too far apart to be considered similar. The issue is that a factor of 1/2 is missing from the pdf definition. To match the likelihood the pdf needs to be
pdf[x_] :=
Sqrt[(1 - \[Rho])/\[Rho]]
Exp[(inN[x]^2 - ((Sqrt[1 - \[Rho]] inN[x] - inN[PD])/
Sqrt[\[Rho]])^2)/2];
Now the result will be
{PD -> 0.013438593418697627473, \[Rho] -> 0.10995316859509961784}
which matches
{137.731, {PD -> 0.0134386, \[Rho] -> 0.109953}}
