Wow, there is a lot of ignorance on this thread. Of course you make a lower hourly rate as a live in. The live in accommodations are part of the compensation.
You are paid adequately for what you do, OP, but it sounds like you have mismatched expectations with your employer. Since you like them, I'd consider giving generous notice, but it looks like you need and want a different job.
Let them know when you are at the stage of accepting a new job.
Would you consider live-in digs an attractive perk at your job? If you lived in the slums, perhaps. Many nannies do come from upper middle class, so your assumption is erroneous. Plus, most prefer a little space away from work and can afford it.
I honestly can't decide if you are extremely self centered or purposefully obtuse.
It doesn't matter whether you or I consider "live-in digs" an attractive perk. The fact is it is, objectively speaking, a form of compensation. Like contributions to health insurance. Like PTO. All of these benefits have a relative worth in the market place and are considered as part of overall compensation. The market value of rent for a live in space functionally affects the hourly rate by depressing it to accommodate the "free rent". The idea that a live in should make the same or more as a live out is basically suggesting that the NF should throw in free housing (and maybe food) just...because. That makes as much sense as saying a nanny should just throw in $800/mo of extra babysitting just because. Nannies should not be expected to give freebies, and neither should parents.
If a nanny wants time and space away from work, awesome. Live out. No problem. Who cares whether you are "upper middle class"? Doesn't matter. You want to live out, live out.
If a nanny wants to live in and not pay rent offsite, she will likely earn a lower hourly rate to accommodate her "live in digs" as part of her compensation.
It's really pretty simple.