I have seen "engine skirt" and "engine nozzle" both used for the bell-shaped end of a rocket engine. Is there a difference?
The terminology is even less rigorous than that! Skirt, nozzle, and bell can informally refer to the same thing.
Bell seems to be shorthand for bell nozzle, a common shape for a nozzle.
One chapter in a book (preview) confusingly does not distinguish between skirt, skirts, and skirt structures. "Skirts" suggests Victorian hoop skirts, which indeed look like the loud end of a rocket. But in clothing at least the plural usage has become archaic, so that may be where the singular "skirt" came from.
Two special cases distinguish nozzle from skirt, though:
An expanding nozzle has two skirts. Here the nozzle is (as usual) the structure that converts high pressure gas into high speed gas, but it does so by means of two concentric skirts, one at a time.
A plug nozzle as in a garden hose sprayer again converts pressure into speed, but with a blockage in the middle of the flow. So here the nozzle is the entire structure, while the skirt is just the outer wall, excluding the central blockage.
Lijat's answer is a third example of a skirt being part of a nozzle.
In this article they use skirt to describe a nozzel extension for an upper stage engine. https://arc.aiaa.org/doi/abs/10.2514/3.57282?journalCode=jsr