What is the proper procedure for a piston engine fire on the ground (e.g. during engine starting)?
As you said, the &quot;proper&quot; procedure is whatever is in the aircraft's operating handbook. Having said that, there is a general pattern, at least based on a few POHs I have to hand. Note that they all list the procedure as &quot;engine fire on startup&quot;; there's no general &quot;engine fire on the ground&quot; procedure.
The 'generic' steps are:

Continue cranking as usual to pull the fire and fuel into the engine
If the engine starts: let it run for a few minutes then shut down as normal and inspect the aircraft for damage
If the engine doesn't start: open the throttle, cut off the fuel (mixture, fuel pump, and/or fuel cutoff as applicable), and evacuate the aircraft

Here's an example from a Cessna 172S POH (Lycoming IO-360-L2A engine); bold items are memory items:


Ignition Switch -- START, Continue Cranking to get a start which would suck the flames and accumulated fuel into the engine.

If engine starts:

Power -- 1800 RPM for a few minutes.
Engine -- SHUTDOWN and inspect for damage.

If engine fails to start:

Throttle -- FULL OPEN.
Mixture -- IDLE CUT OFF.
Cranking -- CONTINUE.
Fuel Shutoff Valve -- OFF (Pull Full Out).
Auxiliary Fuel Pump Switch -- OFF.
Fire Extinguisher -- ACTIVATE.
Engine -- SECURE.   a. Master Switch -- OFF.   b. Ignition Switch -- OFF.
Parking Brake -- RELEASE.
Airplane -- EVACUATE.
Fire -- EXTINGUISH using fire extinguisher, wool blanket, or dirt.
Fire Damage -- INSPECT, repair damage or replace damaged components or wiring before conducting another flight.