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.

there's normally at least 1 level below a written warning which is quite a serious thing