In my experience the first date is mostly about getting to know each other and seeing if you click. Whereas the second date can more just be about having fun together and escalating things further. So I like doing something where the attention is off of the two of you. Something like seeing a comedy show, playing mini golf, seeing a movie.
Depending how far youve escalated things, I would also be keeping it convenient for you to invite her over to your place. Once there things can escalate quickly.
If it works for you thats great but it seems unnecessary and kind of gimmicky. If her justification of “Why would they be doing something for you, unless they liked you?” is correct. Its already taking place. Why did she give you her phone number unless she liked you? Why is she meeting you for a date unless she liked you?