I only watched a minute or so of the first video posted, so this is based on what I saw.
Like Deserthorse said, braced legs and arms. When you brace for a stop you will get the same out of the horse. Relax, and drop your ribcage, LIFT your hand with a curb bit( if you still insist on using one) if you want to send a signal. Do not try to set the head by pulling the head down-it is counterproductive. Again like Deserthorse said lots of circles and stop on the bend, and instead of asking for the back up try reversing using the bend you stopped on. She will already be setup for it. On the bend it is harder for a horse to brace rather than going straight and asking for backing. Horses that have the habit of bracing get sticky there.
If you rode cutting horses you should know that when cutting the rib and shoulder is lifted up away from your cowside leg. If not, he falls into the turn losing ground and time. So think of it that way. Stopping with a slight bend and switching directions.