What are the metrics of observation conditions? How do professional astronomers measure and describe (ambient) observation conditions?

I know smog, light contamination from the city lights, fog, clouds, rain/snow, humidity etc all contribute negatively to quality of the image, obscuring the celestial objects more or less, but I'm fairly sure there is a more precise measurements of viewing capability for any given location or time than "good weather and far from cities" or "Kinda can see M42 in Orion if you squint."
I'm not an observer, but there are a couple of quantities I know:


seeing: is a measure of atmospheric turbulence and airmass. Its units are arcseconds and is the measured size of a point source, typically a star, as it appears in the image. In short: atmospheric turbulences shift slightly the position of the star, so if you observed it for long enought, the star will appear blurred. Good places for telescopes have seeing of few arcsecond; Excellent ones reach about 0.6 arcsec. This link can give some more info
fog, cloud, rain, snow: you don't even bother opening the telescope domes as you won't be able to observe much
humidity: I think that part enter in the seeing. If you do just imaging what you get is some extra blurring of the image. If you do spectroscopy you will see a lot of atmospheric lines (due atoms and molecules, e.g. water, in the atmosphere) that you want to remove from the measured spectrum.