On my generator panel, there are four wires that connect to the outside world.
One is ground, which connects to the start switch common connection (middle terminal), the "running" light inside the start switch, and the hour meter. All three are jumpered together. (The "running" light has two terminals that are physically on the start switch body.)
One is the start line, connected to the start side of the switch, which is shorted to ground to start the generator. This presumably goes to the starter solenoid (and control electronics), and I suspect would read as +12V with a multimeter for that reason. It's not a direct connection to +12V.
One is the stop line, connected to the stop side of the switch, which is shorted to ground to stop the generator. It also would go to the generator control circuitry. It may not show voltage when the generator is not running, and even then might not be a steady or standard 12V.
The fourth wire is a 12V "generator on" feed that connects to and powers the light and the hourmeter, which are jumpered together in parallel.
If I'm understanding your post properly, it sounds like the same setup, with a missing connection to the hourmeter and possibly light. The ground side of them should be connected to the ground on the switch if indeed it is the same circuitry. If it's a different setup, then--well, it might be different.