Originally 4K came from the DCI spec for digital cinema and it was 4096x2160. For consumer use it was shrinked to UHD which is the 3840x2160, but cinemas still use 4K (although most of the screens actually use 2K)

Now 4K refers to image roughly the width of 4000 pixels according to wikipedia [1]

[1] https://en.m.wikipedia.org/wiki/4K_resolution

All new/upgraded auditoriums installed 4K (laser) projectors when I left the business 5 years ago. So not sure if most auditoriums only have 2K these days?

IMAX has only started switching to 4K laser projectors a few years ago, they were one of the last hold outs for 2K projectors (and many IMAX cinemas still use 2K projectors sadly).

Are those cinema projectors or more like video projectors? The cinema ones are more heavy duty and not changed so often.

Cinema projectors. It should be said that I worked in the business in Norway which had converted all auditoriums to digital in 2010, so many theatres were looking at upgrading their projectors around the time I left for something else.

...and 3840x2160 is roughly 4000x2000, thus "4K".

It's close enough for the marketing department, and now that's what it's understood to mean.