The issue isn't about the dishwasher, or the washing. We pretty much all think our way is the best way of doing things. The issue is about communication and compromise.
Can you sit down and ask him what exactly he feels when someone does something that isn't the way he'd do it?
Is it anger? Does it upset him? Does he feel disrespected? Does he want to help try and fix it?
Or is there something bigger going on? I snapped at my partner the other day as a pan from the night before hasn't been washed up properly.
I didn't snap because of the pan, I was upset because I was tired and exhausted and I'd felt like he's not been pulling his weight over the last couple of weeks, because of that I had been a bit of a martyr so was being an utter perfectionist about the house so the moment I saw something not done properly (which was totally accidental on his part), I snapped! In reality I hadn't had the conversation with him to say 'hey, I am struggling right now, I need you to take on more'
What I'm trying to get at is there a bigger issue underneath the surface? Has your relationship changed with having kids? Are you as close as before? Is he resentful for deciding to take on the house admin?
I think this is a case of both of you don't know how to handle this, so you're in a cycle of he gets frustrated and criticises you, that frustrates you and you clap back - I don't think that's gender specific, I think that's a very common relationship dynamic!
It might be that he's just a controlling dick. Maybe that's the case. But it's clear that there is a hell of a lot of resentment building, which is a relationship killer. Would some couples therapy be something you'd both be willing to try?