I think it's probably a date. I don't think people really have "dates" nowadays. People do this "safer", "half-way between date and hangout" stuff, so there's less pressure and it's less awkward if you decide to just be friends. Which is cool, I guess. It's all good. Just see what happens.
I guess she will - its just you two and you are paying == but ts good to take things slow and just see it goes.
If you do want to offer another time then kinda think you should be upfront - or you maybe end up friend zoned and wanting more . Honesty the best way.
If I was her, I would only think it was a hangout because you said "hangout". I would just take things slow and sort of scan things out. If it seems like she is having a good time, laughing with you, smiling a lot, and you two are enjoying each others company, then ask her out for an official date at a later time good luck!