Will I be a Nanny/Baby-sitter or a Tutor?

I'm starting a new position with a family, two days a week (this is in addition to the 3 days a week with another family). I will be taking care of their two kids, ages 8 and 10. In addition to the normal babysitting/nanny care I will be helping them with their schoolwork. They do the virtual academy program (public school online). The older child does his mostly on his own but the younger girl needs me to go through most everything with her.

My question is: for tax purposes, does this make me a nanny/babysitter or a tutor? And does it even matter?

I wan to do this the legal way and everything I've found online says that a nanny/babysitter is a "household employee" and is subject to nanny taxes. It isn't so clear for a tutor. I've found very little info. I will get paid about the same no matter what, no matter whether the family pays taxes or I take care of it as an "independent contractor", they will just adjust my hourly rate to make it fair. They'd rather not have to deal with the paperwork to be "employers" and I don't blame them.

Any advice?