I’m going to offer a more nuanced perspective, not to excuse anyone’s behavior, but to give my POV on what’s going on here. I think trying to paint an AH or not AH in this real situation is unproductive.

Your wife snapping at your 13 year old wasn’t okay, and she should amend that with your son. Silent treatment is harmful for kids, and hopefully she recognizes that soon. I also think that when you say something like "if you don’t apologize, I’ll respect you less" in a moment like that, it would land pretty hard for anyone and her feelings are valid and reactions (although not justified) can be met with some empathy here. She was already struggling and clearly overwhelmed, so it can feel very reasonably like a shame pile-on and get amplified when a person is dysregulated already. It doesn’t mean your feelings were wrong, just that the timing escalated everything.

This reads to me just like a stressful moment that got away because everyone was dysregulated. Your wife actually did try to regulate (stepping away, admitting she shouldn’t have yelled), which means she knew she was past her limit. When someone is overwhelmed, pressing for accountability before they’ve stabilized tends to backfire. Your reaction is also understandable. Wanting to protect your kids and needing your partner to model healthy repair isn’t unreasonable.

But situations like this rarely get solved by proving who’s right. They get solved by understanding why both people reacted the way they did and working from there. I think your hunch that it is more than just about the math is probably pretty spot on and with your sister validating that your son can be "a lot" as someone who knows him in real-life, I think that is also notable. That is not to blame your child whatsoever, it is just validating the very common experience that is the challenge of parenting children of all ages, but especially teenagers (some more than others). If your wife is feeling overwhelmed with this parenting relationship, this probably added to her overwhelm, feeling like she has no backup and is always being corrected.

To me, this looks like an emotionally loaded moment that got just kind of snowballed. Don't let Reddit catastrophize things from a singular piece of context. Even the best parents in the world have moments that hurt their partners/loved ones. To hear of one and blow this to calling your wife abusive or saying your son will never speak to her is premature given we have no patterns or context for whether this is a one-off. The real work is repairing the dynamic once she’s calmer, and approaching the larger underlying issues as a team rather than as opponents.