I just finished replacing my shifter **** assy...basically, pulling out the button disengages the shifter **** assy from the shifter and once this is accomplished, the only thing holding the shifter **** assy on is the plastic 4-tab compression connection (shown in 7th "step" above)...this is what worked for me:
Leaving the shifter in "Park" position, I was able to get more space between button and shifter when pulling button out. Just enough space that I slid the Red "tool" (that came with the new one), into the created space and with a little pulling out of button, was able to get the button to "travel" freely like the NEW shifter assy. If you look down into the NEW shifter assy, you will see a rod/pin that pulls up(actually down when in operational position) and out of view when pulling button out from assy, essentially disengaging shifter **** from shifter. With some extra "up" pressure on shifter **** assy, the assy came off intact. I installed the NEW one in reverse order and had a little rougher time than expected reengaging the button to operate correctly, but it eventually pressed into place. My educated guess is that you need to (while pressing in on button) carefully/slightly pull up on shifter **** assy to help maybe increase some space for the rod/pin to get pushed back down into proper position in shifter.