So I've pondered this a bit more.
It strikes me that all the stuff that happens in the first film is a bit crap for a game unless you can make it about trying to escape. Apart from the very end, the T-800 is indestructible, and strolls through a police station being shot multiple times but barely even slows down. I suppose the police station bit might be some sort of game - throw enough police at him to allow the heroes to get away.
However! In T2 (which is one of the best films ever made despite the lunacy being espoused by Mike earlier in this thread), there are actual fights between the T-800 and T-1000, and although the T-1000 always gets back up, it can be temporarily dismantled and have to rebuild itself.
So! T2 seems like a better bet for some sort of game. Which means the antagonist is a T-1000. Which means, as Pete pointed out earlier in this thread, it can be any model...