From my standpoint (very busy airports in California), "[Callsign], engine fire, stand by" is plenty for the first call. Just remember that the thing that's going to be problematic for me is the question of determining with which positions/sectors/facilities I'll need to coordinate, as well as terrain. If you're off the 24s at LAX, are you needing to turn left to cross the 25 departures? Are you going to turn hard right and fly through Santa Monica's airspace on the way to your close encounter with the mountains? Are you going to swing around to the right downwind and get into Santa Monica's rwy 21 left downwind on your way into the LAX arrival controller's airspace? If I've already cleared you to, say 13,000, are you going to go up and fence with the arrivals entering the downwind, or turn just enough to enter a center sector different from the one that's expecting you?
In short, my primary goal is in ensuring that you can do what you have to do. I just need to know where to get everybody else out of the way as soon as you've got a moment. Buy hey, if you're on fire and I'm not, I can wait. I'll work it out.
Dave