I don't think I did anything hostile, I got a rude email and replied with, basically, "that tone is inappropriate". The customer later apologized and we walked through his problem and fix together.

Yes: in our business we understand that customers might have a bad day. Maybe even drunk. So we always answer very politely - next business day. If we get rude answer then oh well... what is the point?

Good idea by you to wait a day, so the for the moment angry person has had time to calm down