@Fanwithoutaguard I agree wholeheartedly with everything you said.
I live in a country where having staff (other than a fortnightly cleaner) is quite shocking, very little understood, and open to only the top 3% of rich people. Even those rich people play it down, and tie themselves in knots trying to publically justify how egalitarian and necessary it is.
My family live in another country, where you barely have to crack ‘upper middle class’ to routinely have 2 or 3 staff, one practically live-in.
I’ve learned that there is no point explaining one culture to the other. None at all. ‘No staff’ cultures will never accept or understand that giving people jobs is preferable to doing your own ironing in some kind of egalitarian protest (that nobody notices or cares about).
That said, my ‘having staff is normal’ family would advise you (OP) to give up this nanny, pronto and with true regret. Those two things aren’t mutually exclusive. No one has to be labelled a monster in this scenario (and hundreds like it). Not her for ‘pushing it’, not you for firing her.
You have to fire her because you caught her. Not because she did it. Catching her in lies but keeping her on sets a precedent about lying. A precedent she won’t be able to resist pushing further, and you won’t be able to resist checking up on. She’s employed to take stress and work off your plate for a fee. If she keeps taking her fee, but has now given you extra work (worry, supervision, new policies, Plan B for the dog, etc) then the deal has changed irrevocably. And has no value.
Trust me that had you been less concerned with being nice, ‘rich person guilt’ and treating her as family, she would not have contravened the unwritten rules, and she’d now still have a safe job.
Create some distance with your next nanny, for both your sakes. It’s not at all easy to achieve, but ‘when in rome’ actually tends to work better than we assume.