she is new. you can give her what you think is fair for someone you are TRYING OUT. I would not commit to offering vacation or sick leave until sh works for you two months. It gives her incentive to do her best if she wants to keep the job and then you can eases. if you decide she is worth it, give her what she asks; if she falls short, you can't take back what you have already given you can only fire her.
keep the agreement as simple as possible to avoid confusion

You seem to take issue with her approach more so than the actual details of her expectations, and I think you should consider if she really is a good fit for you. I think her approach (maybe not her phrasing, I wasn't there) is one of a confident professional nanny, who knows she can't sit around and hope you give her what she needs. You seem like you'd like to be in more a position of authority, call the shots, and get to play big boss and that is why you don't like her telling you what her terms are. If you're honest with yourself, and you want a more deferential nanny, perhaps she is not right for you. You will drive each other crazy in a stupid power struggle.

I like this poster. My boss is like this and I just have a ton of patience.

Part time with guaranteed hours, or part time where you pay her if you use her?

Her requests are legitimate. You are not so smart if you think you have to pay her for 40 hours per week during vacation when she normally works 25.