Dude - you just need to become more familiar with the movement. You're trying to re-teach your body something it's forgotten how to do, namely work the hips and ankles rather than your knees and back. Sit back and listen to icecold sooth you...
You're still doing a high-bar squat there, which is fine, but you're clearly thinking far too hard about it.
Your core seemed a lot more stable during that video, regardless - you just need to learn how to keep this tight whilst working eccentric glute/hamstring movements under tension. Yes, you are still getting some extension in the upper bar and some bum tuck, but it's all learning.
Low-bar won't magically make these problems go away, in fact they may some of them worse if you don't control them properly (in ways that would apply equally to high bar...). Yes, you don't need as much ankle flexibility for low bar, and you can get away with less hip flexibility with it, but core stability is possibly more important and knee control/glute med strength is too.
