Halt halt makes more sense ;)
There is a big thread on half halts here in the Dressage subforum, if you scroll through a couple of pages you'll find a very long thread started by "Spyder" - which will give you a lot of information on half halts, their uses and applications.
A half halt in its earliest stages can be ridden somewhat how your instructor is teaching. It is basically a small 'check' to rebalance the horse in order to rebalance the horse over its haunches, to prepare it for the next movement, and to check the response to your aids.
It starts out as a 'nearly' downward transition. Gradually it becomes so refined that inexperienced onlookers cannot see the rider or the horse, using a half halt. It should be a nearly imperceptible shifting of the horses weight more to its haunches.
Generally they are not ridden 'stand alone' but in a series. During a test, I would use hundreds of them, even in a basic training test.