Good question, I wish I were more certain of the answer. As far as reporting the incident, we simply fill out the irregularity report, submit to our Chief Pilot, and from there, they make all the necessary notifications. But you are absolutely correct, wrt how the situation is handled while airborne, it is treated exactly as if there WERE a fire. The checklist in fact is entitled; Smoke/Fire/Fumes in the Cockpit. You can see it makes no distinction between the three conditions, you treat them all the same.
As it turned out, our condition was a gasper fan motor that decided to take an unscheduled early retirement. Really no big deal as the checklist directs you in one of the first troubleshooting steps to turn it off and see if the smoke clears. We did and it did. Even still, it was pedal to the metal until we were safely on the ground, if you get my drift.