"One has continuity to the emergency switch, the other NOT"
See if there is voltage on the wire that has continuity to the switch when the engine is running.
I am 99% certain you do not have a BIRD. You most likely have a simple solenoid system. When connected to shoreline your engine batteries probably do NOT get charged. The solenoid you are working with is activated by the emergency switch of course but there is probably a parallel wire (perhaps purple), that supplies voltage to the same post on the solenoid when the engine is running. It may even work with the key in the on position and the engine not running.
The solenoid is grounded via its metal case if only one wire is needed to make it function. If you were testing things at any time and did not have it mounted you could get confused easily. Or if there is rust preventing it from contacting ground. In some cases one of the little posts is also connected to the solenoid housing for ease of supplying a ground.
It is NOT uncommon for earlier model RV's to not be setup to charge the engine batteries when on shoreline. Once you get this sorted we can offer many inexpensive devices to accomplish that task.
