The Solenoid's trigger circuit controls whether the house batteries assist engine starting, or not.
Some vehicles deactivate the HVAC blower during engine starting or the headlights,or power windows, ect, but have it/them On, with Key to on, not start, so turning the key to on.... the solenoid closes, then opens when the starter is cranking, then closes right after the engine starts.
I'd prefer this, opposed to allowing house batteries to assist engine battery through solenoid during engine starting, but it cycles the magnet and slams and separates contacts twice as much within the solenoid, wearing it out faster. I also would not want solenoid passing starter current from house to starter battery, as that will also wear out solenoid contacts faster, and when these fail often the contacts fuse together leading to full time in parallel situation, which then cycles the starting battery with house loads.
The Ideal, in my opinion, Solenoid trigger circuit would close the solenoid contacts only after the engine starts. I often recommend people put an illuminated switch inline on the ignition activated trigger circuit, thus have the option of letting belts and engine warm up before tasking the alternator and belts with the 50+ amp loads depleted batteries can ask for. Some just use the illuminated switch and have the switch always powered live, connected to house or engine battery, that way if engine battery is dead they can flip the switch and house batteries assist, or if depleted batteries cause their belts to squeal on a cold damp morning, they can turn it off and allow things to warm before trying to feed depleted house bank.
The absolutely full starter battery right after starting on my rig, will briefly draw 90 amps and this quickly tapers back down to 0.5 or less in 45 seconds or so, and if I had a large depleted house battery bank they would be sucking huge amounts of amperage from the alternator at that point too.
I believe a delay in activating the house battery charging solenoid is wise. Less stress on a cold engine whose cold oil is too thick to circulate properly and a damp cold belt might more easily slip and glaze and perhaps cause issues down the road. Each 25 amps the alternator has to make requires one Engine HP, with all the losses through belts/pulleys/diodes, so I'd prefer the engine idle for a bit before asking it to feed house depleted batteries.
I've found a badly oxidized 12v quick terminal under my dash that is not live with KEY to ON or Start, but becomes live only after engine actually starts. After deoxidizing it, I use it for powering my aftermarket voltage regulator, but would use this circuit to trigger a solenoid, if I used a Solenoid and I'd have an illuminated switch on it to neuter it too.
