Question about rocket engine nozzles

[FishInferno]

I think I understand how this works, I just want to be sure I have it right.

 

A rocket engine nozzle or bell is designed to capture the sideways expansion of the exhaust and redirect it down, providing more thrust than a simple hole with gas shooting out.

When a nozzle is underexpanded, the gas continues to expand after exiting the bell, and thus the engine is less efficient because it is not capturing all the exhaust.

When a nozzle is overexpanded, the gas does not expand all the way and fill the nozzle due to high ambient pressure, and thus the engine is less efficient because the unused nozzle is essentially dead mass.

An engine is most efficient when the gases expand to perfectly fill the nozzle, so the engine is capturing all the possible thrust it can.

 

Is this correct? I feel like there are more consequences to an engine being overexpanded than just what I listed.

[Laie]

3 hours ago, FishInferno said:

When a nozzle is overexpanded, the gas does not expand all the way and fill the nozzle due to high ambient pressure, and thus the engine is less efficient because the unused nozzle is essentially dead mass.

The gas not filling the nozzle causes a lot of acute problems like several kinds of instability: you don't necessarily get a stable pattern of exhaust inside the nozzle; if it flickers you have uneven distribution of thrust which not only acts as a uncontrolled gimbal, but also poses structural challenges (engine bells don't like being pushed sideways). Uneven heating, too. But most importantly, (something something mach effects) the ISP may become even worse than that of a simple hole.