In the US in the 21st century vernacular (at least in NYC and Bay Area), nanny means child care provider with fairly regular hours, taking care of a smallish number of children (1-5ish).

The phrase "nanny share" is very commonly used. Sometimes the nanny is in their own home, sometimes in [one of] the parents' home.

Babysitter generally means someone who comes occasionally to watch a single family's children.

There are obviously fuzzy lines separating the two terms.