I puzzled over this for a while, lol, because on one hand, you're right - sit seems like it should mean "do this until I say otherwise" But to me, sit is a precursor. It means "sit until you hear further instruction, but if you get up, it's not the end of the world". Further instruction could be anything, a release, a down, a stay... Adding a stay means "sit until I tell you to get up." If I'm in class, and I'm just hanging around talking with the trainer, and I tell Penny to "sit", I expect a sit. But if she lays down because I've spent half an hour talking to my trainer, no big deal. If I said, "Sit. Stay.", and she downed at any point before I released her, I'd put her right back into the sit.
I guess what I'm trying to say is is that "sit" is just a position. It doesn't imply anything about duration at all. Stay gives duration to the position.