I mean, the commands can mean whatever you want them to, but I use stay more as a "hold that position DONT move a muscle until I release you!" type thing, (although, if he's put in a sit or platz he has to stay regardless if I say stay or not..)
I use wait for when were going out the door, I tell him to wait and he has to wait for me to go and not 'door dash'. I also use it for waiting for me to give him something, waiting for a toy, waiting for anything I guess. Wait is more informal for us.