Maybe we should petition to have this board's name changed to "Frugal Living", given that frugal is a positive word?
Admittedly, my first thought was similar to many - 'nanny' and 'budget'? But
it's a budget issue for the OP, and that's what matters. She - reasonably - expects full value for the money she pays, and is seeking advice on how to achieve that, from us. Works for me!
On a sort of related note - my employer (large company) pays me for the hours I work. They provide me with the tools to record my hours, and they expect those records to be accurate. If they expect me to work until 8, and I leave at 7:15, as an hourly employee there's no reason I should get paid for that last fifteen minutes (frankly, even if my boss tells me 'there's nothing left to do today, go home now').
When I worked retail, just because the next shift's supervisor showed up early didn't mean I could leave before my scheduled time. While a nanny is a much more personalized employee than a clerical worker, it's reasonable not to pay her
if she chooses to leave early.
Oh, OP - if you ever move to Massachusetts, and would hire a nanny with no experience, PM me