What I have noticed watching the first video the OP posted is that the guy in the video at least demonstrates the movement of the legs differently than I have learned from the buttwipe drill:
https://www.youtube.com/watch?v=FWasFdvnGio&t=385s
When you do the drill, you will notice that the back-leg stays straight for quite some time as you slide against the wall and lower your butt, buildung pressure on the instep of the left foot. When you then start bracing by doing a crush the can move with the drive leg, your drive leg will straighten out, wiping your right butt-cheek against the wall while the backleg simultaneously tucks in because your hip turns. You can see this in the Will Schusterick gif playing in the video of the SpinDoctor as well. Schusterick build leverage against the instep of the left foot, when he then starts bracing the left foot lifts off the ground before his back-leg starts to tuck in.The way SpinDoctor demonstrates the move he is squishing the bug as his left leg is still supporting his weight.
What is similar is the way both demonstrate the butt leading the way
https://youtu.be/ZlyD1ynQrh4
Just two things I have noticed, I hope this makes sense without accompanying pics.