The difference between STAY and WAIT
You STAY there until I come back to you
The dog stays in one place without moving until you return back to the dog and release it.
The dog never moves.
You WAIT there until I tell you what to do next
The dog waits in one place until you tell the dog what to do next.
The dog will always move.
STAY is a formal exercise and the body signal to the dog is standing with your side or back to the dog.
WAIT is a less formal exercise and the body signal is to stand face on to the dog.
Examples of when to use WAIT