It was only ever supposed to be two movies (as Camerons original idea was for T2 to be incorporated into T1) there was never any intention to do a T3 etc
Of course, that much is pretty clear. The franchise was handed over to a new studio and new director, as far as anyone else was concerned, it was guaranteed money
I grew up in the 90s and grew up watching the first 2 films. I remember going to see Terminator 3 in theaters when I was 17 and thinking it wasn't that good. I kind of liked the action scenes though. The first 2 are the only ones worth watching a lot of times though.
T2 was the initial script, but due to budget and technology constraints, Cameron decided to write a prequel (T1). If anything, it was supposed to be a one off film.
That being said [outside of whatit was suppose to be], my [franchise] head cannon: T1-T3, T:TSCC, T: Salvation.
You are correct about the 'friendly pet terminator', but it does not negate what I stated previously; and, it is not to say that many changes were made to the 'sequel' script, after the first film.
Cameron probably would have done a future war T3 in late 90s had he not got the Titanic bug, After that he went messiah level ego ..and Terminator was yesterday to him (and then Matrix took Man vs AI action to the next level)