She wants to pay? What does that mean?

It means that she wants to pay for the food next time

she just wants to be equal. and not feel she's depending on you. kind of non old fashion.

She probably is trying to make a good impression on you. If she agreed to go on a second date then she is interested. Women hear all the time about how men hate that women don't at least "make a reach for the check" when we're out on dates. We're told that guys should pay in the beginning but at the same time we hear that guys would like it if we offered. So she's probably trying to show you that she's not the gold digger type and likes you for your company and not your wallet. I would say it's a good thing.

Could be a number of things... It could just be that she wants to be modern and not do the typical guy pays for dates thing... Maybe she wants to set the tone that she's independent and doesn't need you to pay her way...especially if as you said you make more than her...since you are both aware of your difference in income, she may want to set the record straight that it doesn't mean she can't hold her own or that she's not with you cause of what you have... Maybe she really enjoyed the date and wants to do the same for you in return...

She doesn't want you to think she's a free loading good digger. Maybe she's equal..could mean anything. Don't worry about it.

If she was overly persistent on paying for the first date, then you might get a little worried. In that case I'd say she wasnt interested and didn't want to seem like she was taking advantage of you.

On the second date, however, she could be doing one of two things:
A. Offering just to seem nice. (like everyone does in hopes of being shot down)
B. She really wants to pay.

If she insists on paying...id let her. You don't want it to turn into a heated argument.