Relay. Looks like the coil of a relay (solenoid operated valve) that is used to turn on/off the gas supply to the burner. On the other hand it might not be, dont know enough about boiler controls to be sure.
Without looking at the whole circuit I can't be absolutely certain, but I think that it is the operating coil for the relay powering the circuit that feeds the burner controls. That's to say, the controls which check that the pilot light is ignited before opening the gas valve as a minimum safety function. (More complex boilers have forced draught fans feeding the burner as well, although these are rare in domestic usage).
If you look at the circuit then there are 2 ways that the coil can be activated. Either the Central heat thermostat operates, if temperature's below the desired level, or the DHW (direct hot water) relay operates if that function is selected. As a safety/control feature, if the DHW setting is selected then there has to be water flow in the hot water system to activate the bottom relay S2.
Looking at its position in the circuit and the description I would say it serves as a control valve with the purpose to allow the gas valve to fully modulate for load control.
That is to say react to provide exactly the heat input for the required heat output.
Some designs however do not use the mod coil in that way and just have a low and high. These models will not give a DHW output temperature of within +- one degree like the best fully modulating models. I am sure that we will have a boiler expert in the forum who will enlighten us all.
I`m no boiler expert, but I believe the modulator coil has the function of maintaining a constant flame and hence temperature. It should function to control the flame in a linear manner but faults have been reported where the modulator coil appears to `stick` causing the boiler to cycle.
I don`t know what the problem, if any, that WTM is having but hope this may help.
I may have been barking at a completely different kettle full of fish in trying to troubleshoot this pesky thing. Going to study it again, and if I solve it I will post some more detailed explanations here.