How do you like unofficial warning when you did nothing wrong?
"Hey, would you please stop doing that you never did?"
there's normally at least 1 level below a written warning which is quite a serious thing
IMHO failing to avoid engaging in bickering and name-calling is a professional blunder. Just being around and catching the wind means you didn't call yourself out in time. In this case - where nothing sanctionable really happened - a "retraining" that isn't really a warning, just a "gentle reminder", is fair.
I could see it being done generically enough that a person who was wronged sees something being done, a person who did something wrong sees an admonishment, but also someone who didn't actually do anything wrong sees it as a general 'don't be bad' message or thinks it is meant for someone else.
No it's not. It just turns HR departments into ways to anonymously harass people. If the behaviour doesn't merit action then there's no need for HR "training".