Further work on the water heater revealed that the switch to turn it on gets hot if the ignition sequence fails and I don’t turn the heater off. This was decided to be yet another possible cause of the trouble I’ve been having.

I took the switch apart, cleaned the leads, and then reassembled everything. It’s been three days and the water heater has been acting perfectly. Dare I hope the issue is solved? I doubt it will end up being that easy, but I might have bought myself some time.