The only reason a woman will insist on paying for her part of a date is if she is too independent or not that interested and doesnt want to feel obligated.
The only "obligations" a woman has to a man is to be less than five minutes late (unless she calls or text's beforehand). And... at the end of the date (if the man pays) to say "thank you."
I've encountered more women that follow the script of volunteering to pay their half (and then it's my dating cue to say, "I'll take care of this"). In fairness, some of these women may be uncertain how you feel about them... and you paying is a sign of your approval.
The women (and they've been few and far between) that sincerely wanted to pay their half or suggest buying the drinks somewhere else. really impressed me. And those that matured into future dates or a relationship seemed to be a lot more fun to be around (just as feminine, but less hassles and drama).
There are plenty of other reasons that a woman may wish to pay her half.
1. A former date made her feel beholden and she doesn't want to risk this again.
2. She realizes that this isn't the buggy and carriage days and that she has an income too.
3. She's not interested in the man and doesn't want him to feel used.
4. She may know of women that leach and she wants to differentiate herself from that image.
5. She wants to appear less maintenance than the other women that you're seeing.
If you mistakenly ended up taking out a feminist, it's your fault (you didn't screen enough beforehand).