You did what you thought was best for your daughter & that's fine.
im sure your daughter is lovely, but she's 2 & small children don't always report things accurately, it might be THEIR truth, but not an accurate reflection on what happened.
my friend caught her DD as she was at the end of the slide still moving very quickly, I was standing right there,because they were both moving my friend kind of caught her DD's arm and it obviously hurt a bit.
Her DD told anyone who stood still that her Mummy always hurt her, hurt her arm by twisting like this (showing a twisting skin burn I don't know what they're called these days).
her DD wasn't 'lying' is her truth, but it's not THE truth & she was stand off ish with her Mum for the rest of the day, if it had been someone else she'd probably have said she was scared.
I could recount many more examples.
what's done is done this time, but you need to keep it in mind going forward, or you risk going through nannies like water through a sieve then changing schools constantly.