Ditto.+1
You did the right thing OP, the girl you're talking to is just mental. Luckily, not all women share her view!
I did exactly the same thing many years ago. I was with someone, but I met someone else and realised that my feelings weren't anywhere near as strong as I thought they were and broke it off (I've found since that if I'm with the right person then I wouldn't want to cheat anyway). I broke up with them because I realised it wasn't going anywhere, not in order to go out with the other person. I didn't go out with the second person until months later.
I'm fairly sure my boyfriend at the time would have been way more upset if I had physically cheated on him as well as breaking it off. As many others have said thinking about cheating, seriously considering doing it and following through with it are completely different things.