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I think asking her if she knows your name is a bit DLV, slight case of confirmation seeking. Also, it opens up for the kind of answer that you actually got, which is a sure backfire. Instead, I guess, lead the discussion by saying "I bet you don't know my name", which is most likely to lead to you being right (when she says that no, she does not know your name) or her qualifying herself ("yes I do, why don't you think I know your name blah blah").
...that is if you're gonna keep the names discussion up at all. I'd prefer leaving it up to her to ask for your name ( for the case that she, after all, doesn't know it) and just change the subject, thus dominating your exchange.
I completely agree with you in the aspect of wait for her to ask for names however I am almost positive that she knows of me (well I am since she came up to me) and I knew of her because we went to the same high school. So I wanted to get the introduction out of the way because its weird knowing her name without being properly introduced. Should I have said that I already knew her name? I felt that would have sounded creepy so I wanted to pretend to recognize her but not fully know who she is. Also I agree that it was a bit weak to ask if she knew my name but I was curious to see if she would say yes. I'm still kind of confused though of what her intentions were? Do you think I lost her interest that quickly?
Also if I run into her at the bar in the future (which there is probably a high chance of) should I bring up anything about our first encounter? Should I approach her or should I see if she will come up to me again?