Re: She said "I don't believe you". WTF?!
She walked off just like that? Maybe she was just having a bad day.
I usually use "I don't believe you." or "No, you don't." kind of phrases to make girls validate themselves. For example:
HB: I missed you.
Me: No you don't.
HB: Oh yeah? There is this and that blabla...
HB: I'm so blabla.
Me: I don't believe you.
HB: Well I'll show you then...
Of course, I wouldn't use these until further phases of interaction because if she isn't interested in you she could just say something like: "I don't care if you don't believe, it's the truth."
Implementing to your case, maybe she wanted you to validate yourself, maybe wanted you to go after her. I think it was a quick test and if you wanted to take the challenge you could have given a quick response like: "You say that to every person that just wants to talk to you?"
This little tease will indicate that she didn't need to make a big deal out of your approach and possibly calm her down. On the other hand she could say something like: "No, only your kind." which introduces the next challenge. Either way it's now a convo and it should be fun.
Good luck next time.
It's not about who I was or who I'm going to become.
It's about who I am. Do it right here, right now.