I'm really not a TWH person but I do have a Fox Trotter and the head-bob is normal (and desirable) for both breeds. It gives them a rhythm to their gait, kind of like a person swinging their arms when they walk. It helps them get a larger stride.
I don't like the squatting style of the second horse, he is more like a show-ring type horse than the first horse. The first horse looks more like a flat shod, pleasure type horse. The kind you could take out and enjoy on the trails.
What makes them attractive (to me anyway) is that they are fast and smooth. They can out-walk most regular horses and instead of trotting (if you get a good one) you glide! I kind of find gaited horses addictive. I only have one, and am unlikely to be able to afford another (I am not in gaited horse country), but man, my mare is a blast!
She has a 1/2 QH son who shows some promise at gaiting (he has a flat walk) but he is so green that I figure we are years away from working on gaits, I just want him to be a safe trail horse. Gaiting can come after we master the walk-trot-canter and not spooking or bucking.