From what you’ve described, it sounds like she’s a straight forward and direct person, which is good. Just be straight forward and direct (but tactful) with her. Something like, “I’m sorry, I think I misunderstood the situation. I’d like to hang out as friends, but I’m not looking for anything more.” Optionally, “We can have that drink if it’s just as friends, but I’m not going to lead you on if you want something more.”
Remembering birthdays is a reflection of how much someone cares about birthdays. There’s no need to read anything more into that.
Many people (this thread is an example) are either completely indifferent to birthdays or even dislike them. To us (I’m indifferent) birthdays are totally pointless and basically a fact related to identification, like a student number or social insurance number. I remember my student and social insurance number because I have to, in order to function at school and in society. I don’t ask anyone else’s student numbers because it’s entirely pointless and useless information to me.