The Terminator wasn't killing anyone even before he met John
Thats always one issue i had with the movie, when he meets up with John they make this big thing about John teaching him not to kill people and the Terminator doesn't get it at first continuing to say "Why?" everytime John tell's him he can't.
But we saw when he went into the bar he doesn't wanna kill anyone already, he didn't kill anyone at that bar, everything he did to them was just painful but he let them all live, he just broke a guys arm, threw one through a window and tossed that one guy onto the stove. You could just tell he was already not the same Terminator from the first movie who punch straight through a guys stomach and killed all those punks.
Which is why i don't get John having to teach him not to kill, i mean had the Terminator killed all those guys at the bar and looked to be just as vicious as the original but this time only he's there to protect John it would have been fine, John then teaching him not to kill would make sense and show him growing a little.
I know they wanted to show right off the bat that "hey this Terminator is a good guy this time around, see he wont kill people" but that kinda ruins their little arc they try and do later where they have John teach him not to kill, seeing as they already established this Terminator doesn't wanna kill.