That single connector wire...Cubey's photo above, upper left.
That wire connects to the WTS lamp in the instrument cluster. If the key is ON, the instrument cluster is powered up.
Grounding this wire ( the side going into the wiring harness ) should illuminate the WTS lamp. The engine does not need to be running to see this. But the key has to be ON.
The glow plug controller causes the WTS lamp to illuminate by internally grounding this wire.
You can verify that your glow plugs are cycling upon startup by watching your voltmeter and/or any incandescent lights when you are starting up. What you are looking for is the voltage drop on your battery when the glow plug relay switches the glow plugs on and off.
A good set of glow plugs will pull right at 200 amperes from the battery ( about 25 amperes for each plug ). It's effect when switched on and off the battery bus is quite noticeable, as the alternator is not yet spinning enough to bring the voltage regulator into play. This is also sufficient current to exaggerate the effects of corroded connections.
In my van, the WTS lamp is in a little cluster that very conveniently pops out from the front with a little prying along the edge, which makes bulb changes and electrical inspection a lot easier.
I zeroed in on the indicator circuit as you mentioned the glow plugs seemed to be working just fine, but the lamp had a mind of its own. But then mentioned you have to keep your truck plugged in.
What's the outside temperature?
Personally, I think you have either a grounding issue, or problems in the power delivery to the controller.
( Edited to remove crap. Note to self: post THEN have a beer).