The most popular hypothesis is that it derives from French cours, cognate to "course" as in "lecture". 3 Note that the English "cour" is effectively a back-formation from cours (which is singular in French), and the singular/plural distinction between "cour" and "cours" is an English innovation.
The reason Angel Beats seems so rushed is that the show was originally planned for 2 cours (24-26 episodes) before being cut down to 1 cour (13 episodes) late in production, at which point it was too late to re-pace everything correctly for the shorter length.
However, the split took 2 cours, from 29.03.2008 to 05.10.2008. If not for the 3 months condition, I'd say that this was indeed first series to air in split-cour format.
Even if anime season lengths were randomly either 1 or 2 cours, we'd expect to see 50% of two-season anime be "symmetrical". In reality, there is a moderate bias towards 1-cour anime (at least in the present day), so that even absent other factors, we'd expect more than 50% to be "symmetrical".
So a 26 episode series can be referred to as "2 cours". But as far as the length of an anime "season", a 12 episode series run can be referred to as a "season", yet sometimes a 26, or 36, or 201 (Gintama) episode run is called a "season". It's technically not tied to a specific length, which is why some people prefer to use "cours".
It's the same as Japanese Television seasons (or sometimes referred to as "cours"). There are 4 of them each year and each one lasts roughly 13 weeks. The Wikipedia page for Japanese Television Drama says: Japan has four television seasons: Winter (January–March), Spring (April–June), Summer (July–September), and Autumn or Fall (October–December). Some series may start in another month ...
They channel airing the show would just shift things forward or backward 5 minutes. So something that normally starts at 12:00 would start at 12:05. The shows in the evening will start at the usual time by, over the course of the day, removing 1 or 2 adverts each advert break. The channels that keep a strict schedule just remove most of the adverts during an advert break and play the show then ...
Anime releases are also split into cours, 13-week blocks of episodes also sometimes referred to as "seasons". (As described in What is a "cour"? and What defines one "season" of anime?.) Some shows, such as Madoka, Bakemonogatari, and Ore Imo, run one cour in length.
What production in cours brings to the table is mostly stability and risk reduction. If Arifureta is truly a flop, economically, the committee will probably cut their losses and not renew the project for more mixed media projects and move on.
7 According to Japanese Wikipedia, the 3rd season has 2 cours which cover chapter 120-217 (?). The first cour (The Third Plate) covers chapter 120-166: Chapter 120-121 Chapter 122-125 Chapter 125-127 Chapter 128-130 Chapter 131-134 Chapter 135-138 Chapter 139-142 Chapter 143-148 Chapter 148-149, 151-155 Chapter 155-159 Chapter 160-163 Chapter ...
