I only watched the first video.
I like that he shows you to run your hand down the rein as if you are "telegramming" to your horse. Or , in other words, sending a communication to it, and doing it lightly first, then increasing the pressure, so that your horse has a chance to respond on the light cue if it will.
I thought he then went into some rather quick flexions, and the horse got crooked in it's head (rolled the head over instead of keeping it's nose as much on a vertical line as possible. )
In the disengagement, if you turn around and look at the leg you want to step under you may end up becoming off center and unbalancing the horse. If you lean over to try and look at the leg, as some would do, you end up putting your weight into the inside of the circle and this makes it harder for the horse to step his inside hind under and shift his hindquarter over the opposite direction from where the rider is leaning. Horses want to step under where you lean your body weight, not away from it.