Yes, it’s kind of the same thing in T2 since the T-1000 is hunting John Connor, but T2 is more about machine vs. machine, which isn’t nearly as scary as man vs. machine, which could, I don’t know, maybe happen one day?
The first Terminator is mainly a horror movie first, due largely to Linda Hamilton’s and Michael Biehn’s fantastic performances.
The Terminator is about John Conner’s father, Kyle Reese (Beihn) going back in time to protect Sarah Connor from the T-800, played by Arnold Schwarzenegger.
But that makes it a man vs. machine story, which is already terrifying in itself.