House heats up but thermostat is turned off AND thermostat's battery is dead

If all wires are disconnected from the two T - T terminals, when the system is powered up if the relay still activates the L8124C is defective, that was mentioned before.

Are you sure this black wire is not pushing on the armature of the relay (the black plastic part) ??? It is way too close for my comfort.

If it helps you any the relay's coil is visible from the side.

Left - Relay de-energized, contacts open.

Right - Relay energized contacts closed. Pressure from an ill placed wire could push the armature (the black plastic part) and erroneously close a contact.

" Last fall the boiler system was in perfect working order, never a drip, no corrosion. Now there's a puddle on the floor in front of the boiler. How am I going to fix this? Is it fixable? "

DIY or Plumber. Replace what is leaking, may need the tankless coil assembly. Seems like a lot of rust and corrosion for only a few months of leaking.

Is the relay pulled in while the circulator is running without call for heat? Like @109A_5 showed, you can manually press the clapper to see if the contacts disengage. Or you can pull slightly on the clapper to engage the contacts. This thread should be dead by now.

That CK61 is an older version of the one I showed in the diagram above This is the manual for the Field CK-61. The wiring is similar

I have updated the wiring diagram for you to print and place with the heating equipment for future reference. 

@109A_5 has a very good illustration of the 1K2 contact inside the L8124C 1K relay I referenced above.

If that contact is welded together or there is a wire pushing on that armature, then the contacts will always make the circulator operate whenever power is applied to the L1 contact on the L8124C control. That contact in that photo must be open to stop the circulator from operating. If you find that the contact is being manually closed by an obstruction, then removing that obstruction will solve your problem. If the contact is welded together, then you need to replace the control. 

Also @HVACNUT indicates that you can manually move (with your finger) the weight on that armature to open or close that 1K relay to see if the contacts are opening and closing. This will power the C1 terminal and disconnect power to the C1 terminal on the L8124C as you change the position of the armature manually. Once you actually move that part with your finger and see that the C1 terminal stays powered (Defective) or opens and closes (Operating properly) you can determine if you need to replace that control, or not. 

Pushing the weight should stop the circulator and the burner, pulling should start the circulator and the burner.

After looking closely at this statement

Later I did some testing with ChatGPTs directions. For one of the tests I connected C1 and I disconnected one of the TT wires then jumped T T with an insulated needle nose pliers. The relay engaged immediately. I also did a couple of other tests. One for Voltage at B1and B2 both when TT is open and again for when it is closed (jumped); I got 120V when TT was jumped. Same results at C1 and C2. Those results were normal, if I'm following correctly. Yet the circulators are running to heat the house with TT open (one wire disconnected) which is not normal and not even explicable.

I can see that your test may tell me that the L8124C may be operating properly.

When you say you "Jumped T T with an insulated needle nose pliers" and you saw the relay engage and when you removed the jumper the relay disengaged. This is what is supposed to happen.

You further stated that B1 to B2 was powered when the T T was jumped and that B1 to  B2 had no power when not jumped.  This is also proper operation.  

You further stated that the same happened with C1 and C2.  This is also the correct operation of the control.   

I do not understand how the circulator is operating when there is no power to C1 if the only wires from the circulator are connected to C1 and C2.   There must be another wire connected to the circulator for it to get power when there is no power from C1. That other wire will provide power to the circulator and provide power to C1 at the same time. In other words, the power is not coming from the C1 on the L8124C.  The power is coming from somewhere else and powering the circulator that also happens to be connected to C1

After a closer look at your post I am wondering if there is more to look at here. This is what I'm talking about:

"Yet the circulators are running to heat the house with TT open (one wire disconnected) which is not normal and not even explicable."

Is "circulators" a typo, or do you have more than one circulator? If you have more than one circulator, then how many thermostats do you have. There are several ways to connect a circulator based on using multiple thermostats. That multiple thermostat thing can be part of your problem and may have nothing to do with the L8124C control.

So we did some "real time" testing over the phone and have determined that the thermostat is defective. The thermostat wire with the thermostat off the wall, does not complete the circuit from T to T on the L8124C1003. The C1 is not powered when the thermostat wire is connected to the L8124C and the thermostat is off the wall.

When you connect the red and green wire at the room thermostat location the, 1K relay pulls in and the C1 terminal gets power. This is the way it is supposed to operate. The thermostat wire and the L8124C are doing the job as designed.  The thermostat must be defective since the only time the system overheats with the thermostat turned off, and when the thermostat is connected on the wall. Remove the thermostat and everything is operating properly.  

Here is the wiring diagram for your system with the Boiler, Burner with Electronic primary control, L8124C Aquastat, the Field Controls SWG with the CK-61 control kit and a simple T87K thermostat.

You should print this diagram and have it close to the heating system for future reference.

This is some thermostats you may want to use for the replacement Honeywell CT87K or the Honeywell CT31A do not require battery replacement.

You can also purchase other thermostats with more features but the tenant will most likely not benefit from them.