Let me start by stating that I consider chat.se a bit like a free zone. Whatever is considered off-topic in both travel.se or meta is possible on the chat, of course in the boundaries of decency.
However, there has been some face palm behaviour lately between some disgruntled users. Their behaviour is not helping and honestly in this specific case both users are equally responsible, in my opinion.
At the same time I would like to question the sudden massive arrival of external moderators once someone flags a comment as being inappropriate. Usually this is not helping in diffusing the issue, on the contrary.
Yesterday one of these mods suggested to discuss this in meta. At first I wouldn't even consider spending time on it. Chat should remain a lightning rod.
However some users and moderators seem to have the recurring tendency to get carried away once and a while.
So maybe it is indeed time to rule up the chat conversation as well.
For the participants it is easy: BEHAVE and keep it decent. Ignore comments that might look offensive. What is considered offensive in one culture isn't in the other.
For the external moderators I like Dirty flow's comment: "Remove the messages, suspend the user and go on".