Of course it did. That doesn't mean the past was changed. That was always "the past". In other words, it did not construct an alternative timeline. Our story still has one consistent timeline that is not affected by the ability to go into the past and alter past events.
That seems correct, think of the time travel rules in Harry Potter (Prisoner of Azkaban) and Interstellar
All the things they do when they go back in time had already happened. He figures it out in Harry Potter, he says he knew he could cast the patronus to save himself because he had already done it. In Interstellar he writes the co-ordinates of NASA with the dust or whatever, which had already happened at the start.
It's different to Back to the Future where events can actually be signficiantly altered by the time travel. There is an original timeline where he never went to 1955 then he creates a new one where he does.
The Terminator series is just a mess with the time travel rules because at first it leads you to believe there's only one timeline because of the conception of John Connor, he had to go back in time and do it with Sarah because he had already done it, but then later they are able to alter the timeline to stop judgement day