K130... I do trust you understand the difference between flames due a surge (from the intake or jetpipe(s)) and an engine fire - outside the core. The former is to be expected in many engine failure scenarios, and it could be dangerous to get overly excited about it.
As far as Airline Pilot training goes, if a fire is extinquished - as far as the indicatons go - it is extinquished. Emergency probably downgraded from Mayday to a Pan, and sense of priorities altered. I am sure you (and I) could find counter arguments / incidents, but you will have a very strong uphill battle if you want to change this.
In the LAX incident, you can be sure that with any indication a "fire" was present (a call from ATC re flames from an enigne is NOT an "enigne fire"), the aircraft would have been on the deck in minutes, very overweight. Once a fire was judged extinquished, it would be very poor airmanship to land overweight - indeed, a 4 engine aircraft 1 engine out without further complications is not generally held to warrant even a Pan (this all assumes one understands the definitons of Pan and Mayday).
NoD