A battery cutoff is not normally a solenoid, because every time you parked the car, and did NOT kill the battery, the solenoid would be drawing current.
It’s a latching relay. Pulse 12v on one circuit and the relay closes and the car has power, pulse 12v on the other circuit and the relay opens and kills power. No power draw from the relay when it is closed or open, only when changing its state.
