Do different planes (category) aim differently in the 'final stages before flaring'?
Because of the downrange offset between the PAPI and the GS antenna, only a P3 (jumbo jet size) PAPI (14 meters eye to wheels) gives a more or less parallel path between the ILS glide slope and the pilot's eye so that they are both aligned on short final.  The other PAPI categories all have a deviation between the ILS slope angle and the PAPI slope angle, so that if you stay on the ILS glideslope all the way down, at some point the eye height becomes too high for the PAPI and you will see three whites even though you are right on the ILS.

You take this into account in training to learn to resist the urge to dive to correct because the last thing you want is to increase sink rate close to the ground. So really, once you are really close in, inside the airport boundary, you don't rely on the PAPI lights.  If you were on-slope in a stabilized approach configuration at Vref at decision altitude, you just have to hold the pitch attitude and speed with maybe a few small pitch tweaks based on the runway sight picture and you will cross more or less at a wheel height of 50 ft. The goal is to hear "50" from the GPWS just as the threshold is passing below, and the speed tape pointer is right at Vref.  From there the touchdown point takes care of itself using the normal flare/thrust reduction procedure you use for that airplane. 

This Transport Canada Staff Instruction regarding the design of PAPI installations and issues with harmonizing them with ILS approaches gives some useful background: https://www.tc.gc.ca/media/documents/ca-opssvs/AC_302-009.pdf