Should low limit aquastat switch trigger circulation pump?

PGA

Short version of my question: Should the circulation pump be running when my Weil McClain water boiler temperature falls below my Honeywell Aquastat low limit setting of 120, and if not, how can I rewire the aquastat?

Full details: I recently bought a two-story 140-year-old house, with water-filled copper pipe baseboard heating on both floors and a boiler in the basement.

The system: a Carlin EZ-1HPW oil burner, heating a Weil-McClain WTGO-4 boiler, controlled by a Carlin ProX 70200 Controller and a Honeywell L8124 Aquastat with high and low limit settings, presently set at 190 and 120. The circulation pump is a Taco 007-F5-1IFC. There are three heating loops, all on a single zone, with one thermostat on the first floor.

The system heats the house fine when I have the thermostat turned up, but since I'm fixing up the house, I've been keeping the thermostat set at 40 while I'm not there to keep fuel costs down but the pipes still safe from freezing. (Hi-low thermometers around the house confirm that all water pipes are never dropping below 38, even when it's below freezing outside.)

The problem is that when the thermostat is set at 40, the low limit on the aquastat cycles on every 15 minutes or so to maintain 120 inside the boiler. Each time, the circulation pump also comes on, sweeping that 120 water through the rest of the system, where it's not needed, and raising temperatures to 52 on the first floor and 57 on the second. Isn't the purpose of the low limit setting to keep just the boiler at a minimum temperature? Should that pump be running? If not, how can I have it run only when there's a call for heat by the first floor thermostat, but not when the low limit switch triggers heat?

Additional information: The system once heated domestic hot water through a coil in the boiler, but a separate electric hot water heater has been installed, so the coil is no longer needed. For a few days I disabled the low limit switch by pulling the right hand "B" connector on the Honeywell Aquastat, and that worked - house temp stayed at an constant 40 - but as I'd been warned, gaskets inside the boiler shrank and water leaked out of the bottom whenever the furnace fired up, so I reconnected the low limit. Now I have a dry floor but a house heated more than I want. I'm also concerned that the low limit switch will pump unwanted heat through the house in summer, unless I just shut down the furnace for that season.

Photos attached. Thanks!

EBEBRATT-Ed

The low limit is supposed to maintain the boiler water temp at 120 if that is what the low limit control is set for. It should shut the circulator down if the boiler water temp drops below 120. There is probably some differential built into the control which may or may not be adjustable.

The low limit is called by some the "low limit circulator control" because of the two functions it contrrols. It's purpose when you used the tankless heater was to maintain the boiler hot enough to get DHW even when the thermostat is not calling. The low limit shuts off the circ pump on a water temperature drop (killing the heat) so that the boiler can maintain temp for DHW

HVACNUT

Like @EBEBRATT-Ed said, the low limit maintains that setting temperature, regardless of a heat demand, and will open C1 and turn off the circulator if boiler temperature drops below that setting. 
C1 should not be powered just because the boiler is making the low limit differential if there's no heat demand.

Is it possible the circulator isn't running while making differential, but the check valve in the circulator is bad.
Easy enough to replace if it's in the flange. Can't see the volute in the pics but if it's old enough, it could be in the volute. 
Is there any flow check on the supply side. Post some pics from 10 feet away if possible. Floor to ceiling. 

Even if it is a faulty check valve, there are better aquastats out there that offer cold start and condensate protection. 

PGA

First, sorry for the delay. I was traveling and didn't see how quickly you both responded. Thank you for your helpful thoughts and suggestions!

Attached are some photos.

You'll see the return on the left. It has three shutoff valves and the circulator. There are no other valves.

The hot water supply is on the right, extending up from the top of the boiler housing. I see no check valves on it.

To the right of the furnace is the city water supply line, running down from the ceiling. A side loop runs through the boiler from when it used to supply DHW, but now the city water runs straight down, past a shutoff valve, a backflow prevention valve, a pressure regulator and another shutoff valve, before entering (as needed) the boiler at the bottom.

THE CIRCULATOR:

I assume the volute is, or is inside the circulator. As the pictures show, it's a TACO. I don't know if the label can tell you how old it is.

I'm pretty sure I can feel the circulator running whenever the low limit triggers the furnace to burn. I can also feel that heat moving rapidly up the hot water supply line as the circulator pushes it through, and within 15 min it heats up the house (even though there's no call for heat).

1) Am I wrong in guessing this means the circulator must be working? How else would the water circulate so efficiently from the basement through two floors?

2) And if the circulator is running when it shouldn't, would this suggest that I have an improperly wired or faulty aquastat? I've also included some closeups of the wiring,

3) If I were to get a better aquastat offering cold start and condensate protection (recommendations?), what can I do about the water seeping out the bottom of the boiler (presumably from shrunken gaskets) when it cold starts? I'd prefer a cold start system, but presently I keep the low limit connected because the boiler doesn't leak when low temp is maintained at 120. Is gasket replacement something I could do myself? I can take on most DIY projects with the assistance of a YouTube video or two. Alternatively, is it a straightforward job for a service technician?

I see two possible approaches:

Quickest solution: get the circulator to not run when the low limit setting is triggered. That would save a lot of wasted fuel.

Ideal solution: switch to cold start, but get the boiler not to leak at cold starts.

Thanks!

EBEBRATT-Ed

The wiring looks like it is right and the controls are set ok. The way you are set up the pump will run if you are over 120 deg and the thermostat is calling. Check for power to C1 that yellow wire should go to the circulator.

PGA

Thank you all. I figured out that the aquastat (probably) had a bad circulator relay, so I replaced it with a Honeywell L7224 aquastat. Now it's working properly. Circulator comes on at calls for heat, but is not triggered by low limit switch (confirmed using a volt meter at two terminals powering the circulator). I still have heat leakage from the basement to the first part of the living room baseboard loop, and I'm not in love with the idea of burning oil just to keep the boiler at 120 for no obvious purpose. I'd like to switch to cold start, but that causes water seepage from the underside of the boiler, so I assume there are some gaskets to replace. Again, thanks for your time and help.