Thanks for the info, I appreciate it.
The landlord drama is what it is. His goal is to never pay for outside help, mine is to have the heat working with some assurance it's safe. I'm not totally thrilled that one of the failure modes of the relay is to run the heat continuously... I was just out of the country for two weeks, if this would have happened then, that would have been bad. As far as I know the heating units themselves don't have any cutoff. They've been painted over so many times that I can find no indication of a make/model.
If he wants to blame this on the thermostat, that's kind of nuts. It's two wires - close the connection, heat goes on - open the connection, heat goes off. But if playing dumb is the best bet, I can do that.
I will tell him the truth - I had to cut the breaker to stop the heat.
I'm having a hard time dating this place - it's a large standalone 2-car garage with a full second floor that's now an apartment. All the original power outlets are 3-prong, so I think that puts it somewhere after 1970. The unpainted drywall in the garage is tan/brown rather than white or grey. The floor tiles are asbestos, which I think dates it to before the 80's. The heating units look to be part of the original construction - all the wiring for them is properly routed behind the drywall, as is the thermostat wire. I guess my point here is that the relay unit is probably at least 30 years old...