I don't/can't know why you're getting a different result from your CAD software, but I'm pretty convinced that P3_CATIA is incorrect.
One check you can do is to find the perpendicular projection of the different points onto the rotation axis, as with the code below. They should all agree, but I get a significant discrepancy for prj_CATIA
Once you've computed the perpendicular projection it's also easy to verify the angular separation between P3_old and P3new_1, as with the code below. I get 5 degrees to very high precision