Learning my system: WGO-4 firing up-to12x/day in summer

could be the circulator for the indirect isn't running and the boiler is just sitting at high limit and heating the tank by convection, possibly through the return if there is a functioning flow check in the circulator.

There is a problem with your system.  It could be that wiring has been damaged, it could be a defective thermostat for the water heater, it could be incorrect wiring design for the Indirect water heater tank.    What you described to me sounds like the boiler is maintaining a temperature and it does not have the proper control for maintaining temperature.  That boiler is not designed to maintain temperature with the control system you have described. 

Here is what the sequence of operation for your system should be.

1. Start with a cold boiler on a summer day when there is no call for heat. The boiler should be the same temperature as the room the boiler occupies.   
2. The water heater is maintaining a temperature based on the setting of the water heater aquastat. (120°)
3. The water heater temperature drops below the temperature set-point of the aquastat as a result of hot water use, or just because of off cycle loss over several (16 or more) hours of non use.  The aquastat calls for hot water
4. The call for heat reaches the Taco SR 504 (or SR 506) control and the LED indicates that there is a call for DHW (Domestic Hot Water)
5. The DHW circulator operates to move water from the boiler to the DHW tank
6. The oil burner operates to heat the boiler water.
7. The burner continues to operate until 
   1. The call for DHW is satisfied and both the calculator pump and the burner stops.
   2. The high limit on the L8148 is reached and the burner stops but the circulator continues to operate until the call for heat is satisfied.
   3. The burner may restart once the boiler water temperature drops enough to allow the high limit to close contacts. The circulator is still operating.
8. The call for DHW is satisfied and the burner and circulator both stop.
9. The boiler temperature will slowly drop over time and eventually get to the room temperature of the boiler room.  

My first step is to see if the circulator for the DHW is operating.  They are very quiet so I take a straw and use it like a stethoscope and place it on the pump motor and the other end close to my ear to see if I hear anything.  (unless you are in the medical field and you own a real stethoscope)

If the circulator is not operating then I would check the wire that goes from X X on the Taco SR504 relay to the T T on the L8148 aquastat.  If that wire is damaged and connected (often called shorted) then that is the problem and the wire needs to be replaced.  You can test that by removing one of the low voltage wires from the T T on the L8148 aquastat if the burner continues to operate, then the L8148 is defective, if the burner stops operating then the wire is defective or the Taco SR504 X X terminals are defective.  To see if that is the case, I would need to see inside the cover of the Taco SR relay. 

Post a good quality picture of this box with the cover off. What zone indicators are on (if any) when you believe the boiler should not be firing but is.

Is it easy to catch it in the act ? Do you have a multi-meter ?

There are 5 zones that go into the zone panel. Terminals XX in the Zone Panel should connect to TT in the aquastat, with nothing else attached. When the boiler is on, and nothing is calling, are any of the Red lights on at the zone panel?

You say when the burner is on, and there's no heat demand, "the ~180+ heat doesn't run past any zone circulators."

C1 in the aquastat is connected to this circulator. Its got some type of bypass at the back wall. And that circulator is pumping into the expansion tank.

If no Red lights are on at the zone panel, but the burner comes on, see if TT is closed. If yes, go to the zone panel, or there's an intermittent short between TT and XX. If no, bad aquastat. Though I've never seen an L8148 magically aquire a differential.

At the zone panel, there are 5 BX cables feeding the 5 circulators. However, I only see 5 total low voltage wires going into the zone panel. There should be 6. 5 for the zone thermostats, and 1 for the isolated end switch between XX and TT. Something isn't wired correctly.

Are these wires (1 each) connected to the thermostat terminals ? And not shorted inside the L8184A, by design or mistake ? Where does this cable go ? Those wires should go to the XX in the Taco SR504. There seems to be nothing on the XX terminals on the Taco SR504. From here it look like the wires that should go to the XX of the Taco SR504 are just connected together under the Yellow wire nut inside the Taco SR504. I would connect them to the XX terminals of the Taco SR504. If needed move the DHW to zone 4 (thermostat and circulator) and turn the priority on.

It appears you don't have any Zone Valves, systems usually have one method or the other, Zone Valves or zoning using circulators, yours uses circulators. The isolation your documentation requires is provided by the " X-X Isolated End Switch " terminals provided by the Taco SR504. The X-X Isolated End Switch is an isolated contact closure when one of the zones is calling for heat. That isolated contact closure is how the Taco SR504 tells the boiler to fire.

I suspect one of two things why it was wired as found.

Not enough DHW in the Winter so they forced the boiler to maintain 180 degrees with only the High Limit of the L8184A as the boilers temperature control. IMO that wastes a lot of fuel and adds to the wear and tear of the system.

and/or

They did not understand the Zone 4 priority for DHW, so in the Winter time there was insufficient DHW so keeping the boiler water at 180 degrees solved that issue at the price of a lot of fuel use to keep the boiler at 180 degrees constantly.

As mentioned above I would connect the L8184A thermostat terminals (T-T) to the Taco SR504 X-X Isolated End Switch terminals and move the DHW thermostat and circulator wiring to Zone 4 and turn the priority switch on.

Yet another possible scenario. Possibly a troubleshooting attempt that was basically not fully completed.

See image below. Since the L8184A Aquastats have a known issue of the solder joints (on the other side of the circuit board near the Orange arrows) fail where the relay (Red box) is soldered to the circuit board. It often starts out as an intermittent firing situation.

With the L8184A thermostat terminals (T-T) connected together (Yellow wire nut) inside the Taco SR504, if the boiler runs continuously with only the high limit (180 degrees) interrupting the boiler's firing the L8184A's relay solder joints are probably OK.

Since the DHW does not appear to be on Zone 4, the previously posted scenarios may be more plausible, the Taco SR504 was just not installed correctly, so the 'Fix' (for possible insufficient DHW) they came up with was goofy too.

BTW (possible information for the future) should the L8184A's relay solder joints eventually dilapidate, they are easy to repair if you have the tools and skills. However most of the time the Aquastat is just replaced.

Yeah, when installing equipment the installer should find the correct diagrams that best fits the system, otherwise things may not work correctly.

I believe it should look something like this (edited from other drawings). Is there 2 circulators on Zone 1 ? That may shorten the life of that relay.

Also there appears to be a Neutral (white) wire missing (Red arrow), maybe it is under the wire nut (Yellow arrow) and I can't see it, maybe it is connected elsewhere. (You did list this earlier, I missed it) Does that circulator presently connected to Zone 4 work ?

Shut the power off before doing the rework.

I would probably just connect X-X to T-T and swap Zone 3 and Zone 4 (circulator and thermostat wiring), turn the priority on.

Find the missing Neutral and/or find out what's going on with that. Make sure that circulator works correctly.

Then see how it all works.

Also the boiler's circulator is controlled by the L8148A Aquastat, Yellow wire C1 (Red arrow). I suspect the boiler's circulator was running continuously too with the T-T connected under the wire nut. The 180 degree High Limit (the Brown Microswitch connected with the Blue and Red wires) only shuts the burner down not the circulator. When you get it all working as it should be working, your electric bill may go down a bit too.