Everytime you die, you lose some fame, so it might've just been that.
And the broom is indeed only once per ID. I started another character on my account and when I tried starting the broom quest, he gave me the exact same dialogue as my other character that already completed the quest. To make sure, I even started a new account and checked with a character that didn't qualify to start the event, and the dialogue was different, then I trained it and started the quest.
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