I never dated more than one person at a time (not my style), so when I said I was "dating" someone, it basically meant the early stages of a becoming-serious relationship with one person, exclusively. I wasn't the type to do casual dates, either... pretty intense on my part, we're either really into each other, or we're not. I had no patience for guys who danced around the issue and couldn't decide if they liked me or not.
That was my expectation from the other party as well... I would not have been happy "dating" anyone while knowing they were "seeing other people." (Two different things, in my book.) But I'm generally an all-or-nothing person, so that's no surprise
I know that other people use those terms much more interchangeably, as evident in this thread.
After a long time of "dating," the word "together" replaced "dating" in the usual sentences (e.g. "We're together," "We've been together for X months," etc).