I'm on my third jet type now, and all of them have shared this: if you have a fire warning, treat it as real. Shut it down (whether by the fire handle, or in one case by the normal shutdown immediately followed by the fire handle), and if the warning stops before you fire the bottle, great, don't fire the bottle. I don't know of any where you might keep the engine running, even at idle, although I stand to be corrected.
Admittedly, my experience is limited. I've only had one real inflight engine fire, and I never got to see all the parts afterward (as part of the ATS melted its way out the bottom of the engine cowling).
The drive home after an engine fire where neither pax nor crew suffered even the slightest injury is an extremely pleasant experience. I hope this crew enjoyed it as much. They earned it.