Carl R wrote:So, my question is; what happens with my inside foot?
As Max has noted, you are not managing your inside foot with PMTS movements. In the videos above, every turn includes leaning on the inside ski shortly after transition, well before the skis reach the fall line. Since leaning tilts the ski onto its inside edge, it hooks inward. That's what's happening with your inside foot.
Skiing very slowly, with skidding turns; making the inside ski tip grip is what starts the rotation afaik.
Since you've never posted video of skiing slowly, we can't be sure what you're doing. If that's in fact what you're doing, it has nothing to do with PMTS skiing. Why not post some video of your attempts at the Phantom Move with Touch Tilt? On the recommended 12-14m skis? The input you'd get would be very instructive.
Is it a problem in higher speeds?
It's a problem at any speed.
Should I use the inside ski differently?
Only if you want to become a PMTS skier.
Max has given you great insights. HH has given you the necessary movements and drills to learn them. They're all in the books, but you aren't following the books. Nowhere does HH advise PMTS students to "rotate" our skis with "inside ski tip grip". You're inventing your own system.
ToddW offered a great suggestion. Get to a PMTS camp, where HH, Diana and the other coaches will give you drills at which you will fail miserably. (I certainly did. We all do.) Once we realize how limited our skiing actually is, we can start building effective movements that actually work.