Pretty much what DrewE indicated (I was overthinking it).
Normally, the solenoid is not powered and thus the batteries are not connected. Throwing the momentary toggle switch sends 12 V to the solenoid, which links the batteries.
And here is where I was getting messed up. If I turn the ignition one click (radio works, but doesn't start the engine), the solenoid is powered, thus linking the batteries. However, when I turn it one more click (starting the engine), the power to the solenoid turns off. So generally, when the engine is on, the solenoid is off. However, if I then hit the switch when the engine is on, the solenoid turns on and stays on (presumably so that the house battery can charge when I am driving).
So if I am correct, when driving, I can chose to either have the batteries linked or not.
Pretty sure this is right. Maybe. I think.