Not getting 12 volts to coil
With that out of the way, we can probably figure out what you measured.
Remember the question is whether its less than the supplied voltage while the current is flowing. If the battery is the same 11+ Volts with the lights on, then there is not any resistance to that amount of current along that path.
The goal is to find where there are poor connections or bad wires. If it looks bad, or feels loose, just clean it and gently tighten it up if you can.
I think what you're calling J1 is all of the wires that are connected to the welded junction.
With that out of the way, we can probably figure out what you measured.
Even with the photo I don't understand where this femal terminal connected.
Remember the question is whether its less than the supplied voltage while the current is flowing. If the battery is the same 11+ Volts with the lights on, then there is not any resistance to that amount of current along that path.
The goal is to find where there are poor connections or bad wires. If it looks bad, or feels loose, just clean it and gently tighten it up if you can.
That arrangement is one I haven't worked with. Hopefully there is enough slack its not too difficult to deal with.
