The first thing (in hand) is that the horse simple goes forward (raises whip), and stops (drops whip). By using (vertical) hh the horse stays up/open (as in the second vid which is too compressed however), folds the hindlegs, start to quicken, reward. Nothing more. Gradually the handler asks for more whoa than go. Only a few steps, reward the change of balance, the folding of the joints of the hind legs, shortening of base of support by closing the hindlegs under (NOT the forelegs backward and straightened hindlegs like in that first vid). One can also use rein back (because it is a diagonal movement), and transition into piaffe steps (which is also a diagonal movement). Gradually the handler can TOUCH a legs (in halt to solicit a reaction). Reward. And then alternate, to get both hind legs.
However, the horse can be asked to move too quickly with too often a touch (second vid) for too long a period and the regularity is lost. (He does however touch the top of the croup when the horse is too sited. The hindlegs lose their will to step under with too compressed outline (like in that passage). But it is always a playful interaction of active/up/still seeking the hand.
Touching of the forelegs (either in the front or behind) can solicit more of a specific reaction, by it is not banging on them. I disagree that there is nothing to be gained by touching different spots on the leg (for a calculated reaction) in forelegs or hind legs. It can create amplitude (and reach in passage) in much the same way jambettes are use. Those that put the horse between the whip and a bamboo pole sculpt the horse by little reactions. THe stiffer the whip/poll the more for a sensitive horse, the whipper for a dull horse. No whacking on the top of the croup as in the first vid, that is the work of a novice imho.
Work in hand is used so that a horse many start (with simple walk/halt) first when it only wtC. Then it will have an in hand piaffe before the rider sits on it. Then the rider slowly takes over. Unfortunately many horses are never developed in this slow methodical manner, and it becomes about having to shape it more with the rider. And if the rider cannot ride an effect hh, horse much higher/folding hindlegs appropriately/horse ifV/ it is more difficult to start under a rider.
(Realize I have done work in hand for P&P for about 50 years, and schooled with those at the top of their work for this specialty.)