Yes, the small wire that goes to the starting motor is the wire that starts the car. If it's cooked (burnt out) then it will no longer start the car

Agreed on the grounds. On A-series engines, the starter is bolted to the transmission, which is bolted to the engine. The BAT thick ground cable should be bolted to the engine (on a non-painted spot -- usually the timing cover). That's the ground system

On a B310 the S wire (Starter wire circuit) goes from the Ignition Switch 'S' terminal through the dash harness, which connects to the Engine Room harness. There the wire goes to the starter. It also branches off in the harness and goes to the coil side of the Ballast Resistor