Well, or rather, consider the following (Ascention of a Bookworm S3): a 26 episode series airs in fall and winter, making it a two-cour anime; it's not a two-seaon anime because it's actually the third linearally related show.
We could call it the second sequel instead of the third season, but this doesn't seem to happen.
It's been widely used long before Shirobako. But otherwise yeah. It's the subdivision of a season of a show (group of episodes produced together) that fits within a season of television (around 12 weeks). The options in English are too clunky or not specific enough for the context in which people say "cours".
I always get a kick of hearing people hesitate when deciding whether to pronounce it French-like, English-like, or Japanese-like.
The Japanese word comes from French, FWIW. It's also not specific to Anime. It's a term commonly used in the Japanese TV industry. It's kind of fascinating that "cours" is the more seasonal term (as it refers to a period of time matching seasons), and that none of the original terms (クール and 期) are literally related to seasons (as opposed to English where season is used for periods that nowadays have nothing to do with seasons).
“cour” in anime can be distinct from “season” because it sometimes refers specifically to a 13-episode run which airs weekly over the spring/summer/fall/winter, even if two or more cours get logically grouped into a single “season” according to the episode titles.
It wasn't part of some nefarious linguistic plot on my part, it was just a mistake I made out of habit since it's a commonly used word within the context of the thread.