I would run a better background check than through Care.com. Seriously OP? Those checks are so flimsy....They are not even legit in my book.
I believe this nanny is not hiding anything, because if she was she wouldn't be so forthcoming with her information. Care.com is just screwing everything up because they suck.
If the references say she is good, then why not do your own independent background check on this woman? Sure you will have to pay more out of pocket, but it would be so worth it considering your child is involved.