Terminator happened to air a couple days ago
In it, I noticed Reese *does* make mention that he is from a "possible" future. A couple of times. He isn't from "the" future, he's from a possible future. So I'm not going to give you the "variable timeline" argument.
What I did notice, on my viewing, in Terminator was that they had *destroyed* the time travel facility after they sent Reese through. There's no *possible* way that this particular timestream could have sent another terminator or two or three or a dozen through.
I think that's what logically destroys T2, more than your arguments: there *is* no time travel facility, because it was destroyed. If it wasn't destroyed, then this isn't the same timeline and nothing else makes sense.
But T2 is still a fun movie.
