In the first video (I didn't see part where horse moves toward woman), he uses the lariat on the horse's butt because he is trying to help the horse go "through" the anxiety and learn that he can do it and survive. So, yes, he is kind of provoking the horse but he wants the horse to learn that a rope over it, even around it's butt, is acceptable and surviveable. Also, he wants the horse to be able to do this while walking. Notice that the horse tolerates it at first, but he is "frozen", he is putting up with it, but as soon as he starts walking, he cannot hold it in any more and out comes the anxiety and the reaction. The trainer wants the horse to be able to experience anxiety and know that he MAY walk forward and it will be allright.
If you only stay in the place where the horse is never exposed to anxiety producing stiumlae, horse may be fine, but when that day comes when he meets something over that level, he will not know how to just walk through it, he will explode.