The half halt is really about feel and rebalancing and that is why all the elaborate descriptions don’t capture it all that well. There is no formula. The best way to understand the half halt is to feel it.
To feel a half halt, try this simple exercise. Trot several strides, Halt. Trot off. Repeat this until horse and rider are in tune and horse is on the aids and even anticipating the transitions. Then Trot, ALMOST halt and trot off. That feeling, That rebalancing of the horse who is already on the aids and coiling up for a transition is the feeling of the half halt. You can make that almost transiton smaller and smaller but the ‘regrouping’ or 'coiling" feeling is a form of half halt. It can be a half halt just with the seat only or involving more aids.
Any other exercises using transitions such as walk/halt /walk, almost halt/walk for example, will also work to establish feel…