Hi Rick,
You got the numbers correct, your machine is a 2005 production model.
As for wiring diagrams, the Scag diagrams are fairy good compared to some. Most are in color which makes tracing wires much easier.
Red is battery, yellow is charging, green is engine cranking, orange and brown are safety circuits, white is the carb mounted fuel shutoff solenoid, blue is power to the clutch and black is ground.
The electric control module is a simple logic device and when voltage is returned from a safety circuit or more, the requested condition is allowed.
For example;
To crank the engine, the travel levers, parking brake lever, the seat switch, PTO switch in off position must all be either supplying voltage to or preventing voltage to the ECM, with the keyswitch being the primary source of voltage when in the on/engine cranking position.
There are other circuits involved, but function in a similar fashion.
As for the wiring diagram, the yellow wire actually connects to a terminal on the keyswitch. The relay with the green wires is the engine cranking relay. When the safety circuits are in the correct alignment, the ECM will ground the coil in the relay allowing voltage to go to the start solenoid providing high amperage power to the starter motor. In the B&S adapter harness, the black wire is the ground for the start solenoid, the green/white is power from the cranking relay, the red B+ is from the battery positive cable. All the black ground wires are on ring terminals on one of the engine mounting bolts. This area is problematic and may need to be addressed.
I do think that your problem is more fuel and heat related.