I heard back from Amazon. The info on the page is correct. If you borrow a book with your Prime Membership, the notes are saved for later when you return the book. Then.....
They apologized for me not being able to buy the book.

I'm not sure where that came from.
I feel like they copy and pasted in their answer without reading. They didn't answer my question about buying the book and whether it synchs with what you have. So I still don't know if it works the way you think eschwartz.