From my experience, it's never clear cut. I would say, as far as when you're "dating" as in, it's real, that requires an actual conversation like "hey, want to be my girlfriend?". If that isn't stated then you're not really in a relationship, although you may be courting one another. That still means something, as it is in this point that you're really getting to know eachother and build trust.
From your last response there, sounds like she's interested but playing hard to get. Keep trying but never, ever, freak out on her. Sounds like you already did it once... try to refrain. It can be tough if you're really interested but that's the way it goes. You have to pretend, when making plans, like you're nonchalant about it. Busy? That's cool, my friend wanted to see a movie. Free? Great let's hang. The more you push the more she runs...
I once was interested in a very busy girl... a cheerleader lol. We met at work. Went out maybe once... she was just one of those "always on the go" type girls. A year later after moving on and such, I talked to her and asked her about it. She told me she had had a crush on me, so it's hard to figure out why she never made time to hang out (I had asked her out at least 3 or 4 times). I guess the point is, women are odd and upredictable. So long as she's always busy just be a good friend and when you are together, like at dance, be flirty and as intimate as is appropriate. Maybe she has personal/home issues preventing her from dating? Who knows. It sounds like you're on the right track, anyway.
So yeah at this point just be cool, and next time you see her if she's still flirty with you, and she says she's buys, it might be ok to say something like "ok, well I would really like to take out out for dinner, on a date sometime. Maybe you'll be free this weekend?" And if she says she's busy DON"T FREAK OUT.
Oh and it's scientifically proven that she'll be more interested if you touch her arm when you ask her.