I am a pilot and flight instructor with 35 years experience. The Piper PA-31 Navajo is a small twin-engine airplane that carries 6 to 8 people including the pilot. The interior is about the size of a small mini-van. Everyone can see the pilot up front and both engines out the windows. The engines are piston engines, not jets.
One engine experienced a cylinder failure. On an air-cooled engine, the cylinders are individual assemblies attached to the crank case (very different from a car engine, where everything is buried inside the engine block). If one fails, it can be quite spectacular as the burning fuel/air mixture escapes and the now-unbalanced engine begins shaking violently.
1) Shut down the bad engine and feather the propeller (turn the blades into the wind to minimize drag). This causes the propeller to stop spinning. The fuel would also be shut off so as to not feed the fire.
2) Descend as quickly as possible to land, even if not at an airport. Better to land in a field than to burn up in the air. The emergency descent involves pulling the throttle on the good engine back to idle (it is NOT shut down), banking the airplane, and pushing the nose down to get as much airspeed as possible. This creates a spiraling descent. (Descending rapidly in a straight line "unloads" the wings, which could cause severe airframe damage if the wings were suddenly "loaded" by turbulence.) This rapid descent also has the benefit of possibly "blowing out" the fire (much like blowing out a candle).
3) Near the ground, roll out of the dive, level out, let the airplane slow down, and bring the power back up on the good engine to keep the airplane flying. If still burning, land wherever possible and evacuate. If the fire is out, proceed to the nearest airport or suitable landing site.
The passengers were never about to die, and there was no miracle recovery from the dive. The pilot knew the outcome from the start. Every multiengine pilot is trained to do this. It's published in countless manuals and training materials. I teach this to my multiengine students frequently. The "emergency descent" is a wild ride as the view out the front window looks like you're pointing straight towards the ground (really only about 30* nose down) and the airspeed shoots up into the "yellow arc" on the airspeed indicator (the "caution zone").
But Pres. Nelson wasn't describing what the pilot was doing, he was describing the emotional perception of the naive passengers, whose reaction apparently ranged from terrified to hysterical. The passengers perceived that one engine had exploded, that the other had been shut down, that the airplane was falling uncontrolled out of the sky, that they were going to die, and that they miraculously recovered. No embellishment, just a story about an emotional experience. If I put a passenger in the back of an airplane and demonstrated an emergency descent without telling them, they would probably have a similar reaction, even without the spectacular engine failure.
As a side note, for an incident like this, the CAB / FAA / NTSB would not have sent anyone to investigate. They would have merely gathered information from the mechanic who inspected the airplane, the pilot, and the owner/operator. Engine failures in flight, with no injuries or airframe damage, are a dime a dozen and not worth investigating.