Those posteriors look (at least potentially) extremely similar by eye. One just happened to sample a bit further out into the extremely long right-hand tail. As long as the implementation of the ICDF is correct, this should not require a Jacobian adjustment.
The rule of thumb is not whether or not a variable is in transformed parameters; it is whether a parameter defined in either transformed parameters or locally in the model block shows up on the left-hand side of a sampling statement (or equivalent target += statement) that is intended to have the effect of sampling the quantity on the left-hand side from the distribution on the right-hand side. Such a model requires a Jacobian adjustment when the transform is nonlinear (more precisely, such a model always requires a Jacobian adjustment, but if the transform is linear the Jacobian determinant is constant and can safely be ignored by Stan’s sampler).
For more on Jacobians adjustments, some good resources beyond the manual and users guide include this very gentle introduction:
https://jsocolar.github.io/jacobians/
and some of the links therein, especially including this overview of probability theory:
https://betanalpha.github.io/assets/case_studies/probability_theory.html