How do you check? Is there a way to determine if someone has been vetted?

I suspected where so I looked at Tech Ideas-about where they have many videos and I was correct.

 

As far as the switch. My goal was to use a relay (rated to 20A) so that I could use a smaller switch

Use a P-MOSFET such as this, which is cheap, is surface mount, and will handle the total motor current.
It shouldn't require a heatsink as long as the continuous current is ≤10A.

Below is the LTspice simulation of the circuit with a P-MOSFET similar to the suggested.
The cutoff voltage is shown for the 0%, 50%, and 100% position of the U2 adjustment pot giving a cutoff range of 11.5V to 10.4V .
The range can be changed by using a different value for R2.
The 1 megohm resistor R4, provides a small amount of positive feedback to the trip point for hysteresis, so the circuit doesn't dither around the set point.