I don't keep track of how often I break chains. Break with a chain tool, that is; I have never "broken" one in use. I think that "once you punch out a pin" statement is false. This assumes you are using the tool correctly, keeping the pin ends even, making sure you aren't leaving tight links, etc.
I do think mixing brands, as it sounds like you did, is probably a bad idea.
If you are using quick links, my understanding is those aren't supposed to be used multiple times, but honestly I do that too and haven't had a problem. Maybe I'm just lucky or not very strong.