Great photos and details.
The first thing I see is your sporatic voltage readings are showing very, very low voltages. I don't know if you noticed you have the meter on the auto scale selection.
-- The one that shows 105.4, is actually 0.1 volt (that is one tenth of a volt)
-- The one that shows 51.1 is actually 0.05V (or 1/2 of one tenth of a volt)
What I believe you are seeing are each wire going to a circuit that not a direct route to a ground or 12V. In other words the wires are going to an open circuit.
If I am understanding correctly the above readings are showing each side of disconnect solenoid small wire terminals, which would be the pick (energize) circuit. One side should get 12V and the other side should go to ground. What I don't know is, what supplies the 12V and the ground. A switch or a relay could supply either the 12V and/or the ground.
What happens when you press the AUX Battery on switch? Do you see 12V or a true ground on either terminal?
On my 2006 Journey the AUX battery switch is a momentary sw for both the on and off side. So I assume this sw picks or drops a relay somewhere which picks the solenoid.
I am guessing you could have a bad AUX sw or a bad relay.
Note: Zero volts on the meter can be a true ground or just an open wire. To be sure there is a ground, you need to put the + lead of your meter on a known 12V and then use the - lead to touch what you expect to be a ground. If the ground is present you will see 12V on the meter. If the wire is open all you will see is zero or maybe some small voltage, such as the 0.1V reading.
I don't have the time right now to download and research the wiring diagrams to see where the two wires go.
That would be the step, that is to download the wiring diagrams and locate the disconnect solenoid wiring and trace out where the wires go.
The physical location diagram is labeled " Front End wiring installation"
I don't know which of the other diagrams shows the actual point to point wiring (i.e. schematic)