during the evening and possibly some overnight stays.
That would be babysitting in my view. It is not a commitment, you would make a booking each time and they would say yes or no to accepting that booking.
Id possibly be looking for care during school holidays/after school in the future.
That would be nannying in my view. It requires commitment. You are likely to want it to occur on a regularly repeating basis.
Do they spend time getting to know the child/family first?
I meet families first, visiting for maybe an hour or a bit longer, so I get to know the child/children, how well I get along with parents, and get a feel for the location, car parking, ease of travel to/from.
Then I would start doing the occasional evening, which then may become the occasional daytime.
Childcarers will have varying availability, so someone who does evening babysitting may or may not have any availability to do overnight or daytime care.