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
I'm not so much interested in whether some others have a different experience with the current procedures but rather whether your current procedures are exactly as you state. If we had pilots shutting down engines for a fire bell everytime, we would have to add to this a 5% error rate (wrong engine, misconfigured aircraft, etc.) and end up with a far more serious incident than just an inflight shutdown.