well, the first problem is that she's 3. it's a marvelous age for all manner of adventures, discoveries and new skills, but with them come intense new emotions and frustrations. most 3 year olds are 'very spirited.' it's part of the deal.
it actually sounds as if your banishing technique (and i'm not using that pejoratively, it's a valid coping mechanism) is working well. she IS calming down, and fairly quickly.
'i'm sorry, mommy,' is a learned response, not an expression of genuine contrition. we all teach our kids to say it, but then we tend to forget what it actually means. don't hang too much hope on 'sorry' meaning she gets it. it's a rote response.
explaining to her afterwards is understandable but won't do much to help in the moment. she's only 3. she's moved on emotionally from the crisis du moment, and is simply basking in your approval and attention. she's not actually internalizing the lesson you think you're imparting.
it's probably more effective to just catch that flying little fist and say NO and then send her off than to explain and count. NO is a complete sentence always, and especially for 3 year olds. if she hears NO and is removed from your presence, that's the lesson right there.
did the note from the teacher seem to indicate that you should do something about it? i'm assuming that if there are professional teachers on staff, they're handling it well in the moment and simply informing you that something happened. they should do that. it's a good thing.
if they're expecting you to retroactively discipline her, that's another thing altogether. i wouldn't keep in her in a pre-school that doesn't understand 3 year olds.
but i'm betting it's simply a 'this happened, we handled it' notification.
keep calm and forge ahead. frustrating as the 3s are, they're also charming, magical, and over far too soon.
ETA there is something very wrong with 'reality chick.'
khairete
S.