My 2 pennorth for what it is worth.
I have had an engine fire in a B212 which resulted in a ditching. This one had treble indications, a bang from the back, an engine chip warning plus a high ITT, followed by the fire warning. Two shots of the bottle, still indicating a fire, so I ditched. No worries.
I have also suffered engine chip warnings, and shut down an engine, only to relight it for landing.
The generator overheat referred to by SASless indicated, I think, a potential breakup in the generator itself, hence the reason to secure the engine.
The S61 does indeed have a rear view mirror, and I have had spurious warnings. The initial part of the drill calls for the speed select of the affected engine is retarded. This checks for a split combustion chamber, as the fire warning will gradually diminish if the chamber is split, but remain illuminated if either genuine or spurious. You then check for secondary indications.
However, consider this actual scenario, which occurred about 18 months ago to an S61 SAR aircraft operating at low level just off the south coast of UK.
Engine fire, confirmed by warning light and smoke, followed by a loss of one hydraulic system, followed by a main gearbox loss of pressure warning. The crew made land, a couple of miles away, landed and got away safely. The aircraft was a totally burned out. Time from start to finish was 80 SECONDS!! I have done this in the simulator in 90 seconds and I knew what was going to happen.