By all means, do what you think is right. However you have to look at the big picture here. She dumps you, but after SHE'S the one coming to you when she needs something. And she is looking out for herself and yes she is coming to you when she NEEDS something. Even when she is talking to check up on you. She wants to know if you are still there, as a cushion in a way. She doesn't want you going anywhere and moving on without her. That's the situation, in a nutshell.
She felt that when she dumped you, your world would be over and screwed up and that you would be begging after her. Which it doesn't sound like you did. Which is good. I think in a way it made her feel like something was wrong with her, why she wasn't good enough to chase after, even though chasing after her would push her farther away. It would feed her ego, but it wouldn't get you any closer.
However, I don't think hanging out with her and trying to start things up again will get you where you want to be. She called you to talk to you on her birthday because it was a shitty day, she felt alone and she NEEDED somebody to be there. That's why she talked to you. You are so ready to prove to her that you are a new better person, why wouldn't you want to? Sure you are getting mixed messages, every day she is battling the idea of whether she made a mistake breaking up with you, but I'm telling you that as long as you guys aren't in a relationship, she is looking elsewhere. She is keeping you around as a cushion and when she finds somebody else and you are all upset and pissed off, guess what she is going to say? "We weren't really in a relationship." That will be a steel-toed boots kick to the nads when you hear it. I promise you.
That's why no contact is beautiful. You wouldn't be getting mixed messages. If you didn't respond to her B.S. checking up on you and so on, she'd feel forced to say how she felt if she wanted you to stick around. And then you would be getting somewhere. Right now, however, you are just getting her wishy washy, fluctuating emotions, bipolar kind of together but not girlfriend. I guess seeing as how everything played out, you can't just tell her that you need to be on your own and that you need space and THIS IS WHAT SHE WANTED. She wanted to break up with you but she still gets to talk to you and have you when she needs somebody? How messed up is that?
Proceed with caution. I really do hope things work out, but if they don't, you have been warned.
Waking up next to a beautiful girl,
Step outside and say hello to my beautiful world.