I've been running now for over two years. While I don't consider myself a good runner and I'm certainly not a fast runner, I think I qualify as a runner because I truly enjoy running. I look forward to it. I feel good when I finish running and I feel good about the acccomplishment of becoming a runner.
I haven't run any races yet (though I plan to); I don't run the whole time, although I have run three miles without stopping before; I run slow (about a 12 minute mile so far). And yet, I consider myself a runner and I'm proud of that.