Well, there's a very big difference between "Engine fire: some of the combustion chamber's heat and flame has breached containment" and, say, "Engine fire: the engine has exploded, catastrophically damaging your wing which is now visibly on fire". However, both things are reported in the cockpit as ENG FIRE.
There's also a very big difference between "Engine failure: something has damaged or jammed enough components that the turbines are no longer spinning fast enough to produce thrust or drive the generators" and "Engine failure: the engine is no longer attached to the aircraft, which is why it is no longer producing thrust". However, both things are reported in the cockpit as ENG FAIL.
(Un)fortunately, cockpit warnings prioritise the what (so to speak) and not the how or why. On one hand, this makes decision-making a lot simpler for the crew, but on the other...well, in rare cases the lack of insight can exacerbate a disaster. Depending on when exactly the engine gave out, this poor crew might have been doomed either way, but they might have been able to minimise collateral damage if they knew just how badly crippled the aircraft was. And there was a very similar accident to this one (actually involving the predecessor of the MD-11, the DC-10), American Airlines 191 - one of the engines detached from the aircraft, damaging the leading edge of its wing in the process, causing that wing to stall when the crew slowed down below the stall speed of the damaged wing in a bid to climb. If they could have somehow known about the damage, the accident might have been avoided entirely as the crew might have known to keep their speed up.