Turd Ferguson wrote:
... We know he asked the first Sarah Connor if she was, in fact, Sarah Connor before terminating her with extreme prejudice. ...
...He didn't do a sweep of the first Sarah's house to confirm her identity, so it's safe to say he wouldn't ransack Sarah's apartment to do the same (he only does so when she identifies herself on the answering machine).
As you say yourself, at the first house he identified, by asking, that the woman was called Sarah Connor before killing her. No need to do a sweep of the house.
He---it---killed Ginger before ascertaining her identity. So she was still an unknown. It would make sense to then check the apartment to find out who was killed. Assuming Ginger had some sort of ID, or there were indications that another woman lived at the apartment, the Terminator would continued it's mission.
As an aside, T3 stated that Judgement Day 'is inevitable' (which kind of contradicts the whole 'no fate but what you make' business, but so much for consistency). So no matter what Sarah did or who she phoned, Skynet would still be built and wage it's war.