same as space heatersIf you have resistance on the ignition coil then it should be ok, I don't really study the value when checking these. Open circuit isnt good.
I haven't worked on one of those for a while but ignition transformers usually run all the time when on the switch is on the 'Hot' position. There is a flow switch mounted on the pump that controls the fuel flow as well. Its just out of shot in your last picture on the left were the red mark is. There is a magnet inside the housing that should be free to move up when water is flowing. Shown on page 28 of the parts break down. Sometime the springs get damaged and jam the magnet down parts 7 & 8. Or the flow switch can fail.
it shows the coil has a contactor K1Ignition looks like it should be on even when running on cold as its switched with the motor contactor
oh yeah your are right of course never looked far enough down the pageFrom the diagram fault could be with either flow switch, time delay relay or fuel solenoid. As you already said low fuel light works.
K1 is the main motor contactor, aux contacts used for ignition