The way they "train" the horses to "dance" is cruel. The horses have no idea what they're doing, they're just expected to move their feet, and that's all they're "trained" to do. There is no finesse of collection, no rhyme or rhythm to it, it's just ... I don't even know how to put words to it.
To train this movement, the "cowboys" cross-tie the horses in a small space, then wet their backs and haunches down, and whip them to make them "dance." There is no training up to this point, it's just grab and go.
That second video you posted, charriot, I have no idea what those "riders" were trying to make their horses do. The spin at the beginning had absolutely no pivot to it, the horse was just turning around fast. A true spin is when the horse rocks back onto its haunches and moves its forehand around the haunches. That horse was just terrified and running in a circle.
Then that "piaffe" type looking thing.. that was a confused horse that was just moving its feet to keep from being hit. A true piaffe is nothing like that. http://youtube.com/watch?v=BmV987TzK2I&feature=related http://youtube.com/watch?v=wcnfXBMZy-M&feature=related
These horses are frightened, and have no idea what they're doing.. they just know if they "dance" and keep their feet moving, they won't get whipped. http://youtube.com/watch?v=SlS85Peki3o&feature=related
This horse is a little better, but the Piaffe is supposed to be forward motion and impulsion.. not this stuff. Note the stiff hind end. Pretty sure he's getting whipped too.
This is a nice piaffe, note the forward impulsion and correct use of the hindquarter. http://youtube.com/watch?v=TAiIoswv57U&feature=related
This stuff makes my blood boil.