2018 Update to this, from a personal level:

I'm not going to get overly specific, but there were some things going on at the time that prevented me from seeing something that was pretty painfully right in front of us:

We (or at least I) assumed everyone involved was a man, except in the case that the user visibly identified as not being a man. Had we not done that, we would have understood why folks involved continued to get increasingly upset as we continued to apologize for the wrong things.

While this is unfortunately quite late, I take responsibility for this personally, and can only offer the apology I should have offered some time ago.

I know there's an apology right under this, but I essentially stepped on someone's foot and then apologized for setting their couch on fire. Or the reverse of that. In any event, I didn't handle this as well as I could have, and I own that.

I'm sorry.

As I noted as an edit to my previous answer, some mistakes that I can only describe as bone-headed were made on our part in the course of handling a recent situation.

tl;dr; - we failed to realize what was right in front of us, and we mistakenly accused someone that was actively helping the situation cool down of being combative and ill-intended as we floundered in helping a very confused inexperienced user understand what was happening by blaming instead of teaching.

While everything surrounding this chain of events is a bit bizarre, it's our job as the overseers to get stuff like this right. We didn't, which let some people down and hurt them, which means we've got some apologies to make. Let's start there.

satyrwilder was issued a very stern, and completely inappropriate warning for 'ganging up' - this user did nothing more than try their best to defuse the situation while mentioning our code of conduct, and kept reminding people to be nice. She was also not heard when she was trying to tell us that we had / did everything wrong. Julia Anne Jacobs was issued a suspension that was excessive in the context of being used to temporarily freeze the situation so that we could make sense of it. She needed our help as a very confused inexperienced user, and we didn't listen. Moreover, we told her she was doing bad things while she was just repeating what other seasoned users had suggested, which we should have seen. I'm pretty sure Glyph pulled out some hair as I remained dense in light of him trying his best to say I think you've got this completely wrong.

What bothers me the most is that several of us read the chat transcript and we still managed to get it wrong. We looked at every single letter, we just didn't get it.

To the three of you, we owe you an apology, and we hope that you'll accept it. You trusted us to do a much better job than we did, and we let you down this time. You have also been extremely gracious as we worked through this, and we very much appreciate that.

How in the heck did this happen?

Look back at Glyph's original question. Do you see the mention of a post by "Concerned Father"? Yes, a troll happened upon the confrontation as it was progressing, and took it upon themselves to 'get involved' by typing complete rubbish into a text box. Once we ascertained that it had no connection to either party in the argument, we set it aside as we continued to try and figure out what was going on. In doing that we completely negated the fact that people get harassed on the Internet far too regularly, and how the heightened state of anxiety folks could could have been feeling contributed to things.

We then looked at chat which was getting a little hotter near the end, looked at comments, and issued one suspension to stop things from escalating even more while we worked. That's not an uncommon thing for us to do when people are arguing quite heatedly, but what was really needed was a 'side bar' to explain some things about the system in a much nicer way.

To top that off, instead of thanking someone that worked really hard to keep things sane, we sent a very stern and cold warning, because reactions on our part started happening as the chat was being read.

I then took notice, looked into the situation, and only really understood half of it. 'Snow blindness' is the only term I can think of to describe it. 'Dense' equally applies.

What about going forward?

We, the community managers that oversee the workings of community moderation on our sites, must catch things like this, 'perfect storm' or not. We're going to be extending our training for employees on the community team based on what transpired, and look at existing guidance to fill in ambiguities.

We will also be sharing quite a bit of this with our community moderators, through our existing monthly email to them that goes over important things to know.

Is it possible to keep mistakes from happening in a network of our size? No. Is it our job to catch those mistakes with near-perfect accuracy? Yes, and we need to get better at that, which we're going to do - starting right now.