First, thank you, Decimus, I now want to change my screen name to Dollymop.
Back on track... FWIW, I'm about Sara's age and I don't hold the view that a first date must of course, without even the need to speak of it, include dinner, even when other non-dinner activities have been specifically agreed to. Nor do I feel that the man must automatically pay for this dinner, should it occur, just because he's the man.
That said, I don't think it's necessarily wrong of Sara to feel this way. Hey, whatever, she can believe that a true first date must include hamster races and duct tape crafts. What I think is wrong is 1) to not state her expectations upfront when the date was in the planning stages (even better, put them in her profile) and 2) to try and pressure the guy into paying when he clearly had no plan to (first clue--didn't invite her to dinner, second clue--had already eaten, third clue--spent no money on himself at the restaurant, fourth clue--did not reach for the check!).
Other posters have covered 1) pretty well, but I dislike 2) a lot as well. Like I said, I think it's fine if that's what she WANTS. But, on a first date especially, aren't people paying close attention to the other person, trying to figure out if they're compatible with each other? Especially if they've only just met online? I just think it's very rude that she ignored all the signals he was giving off that he was NOT going to pay for dinner, and tried to get him to do it anyway. It makes her seem very self-centered, like she wasn't even interested in his thoughts or beliefs about things. For me there's just something very entitled about one adult saying to another adult, practically a stranger, "Hey, aren't you going to buy this for me?!" when the other person has shown NO interest in doing so up to that point.
If she had turned him away at the door when he said he'd already eaten, I would think her less rude on that score--at least that would mean she realized he was probably not of the same fundamental attitude as her, and respected his time enough to not waste it further. From that point on it just feels to me like she was trying to see how much she could get out of him, and I don't think that's a polite attitude to take on a date.
BTW, OP, I appreciate you withholding the rest of the date story so far. In no way am I criticizing those who are curious, but I think this has been a really interesting discussion so far, and I have a feeling that if you DO post the rest of the date, the discussion is going to change completely!