I guess for the same reason we use the French word "romance" for "a book about love" - because the Japanese pioneered adult visual novels, porn games, etc. When the West only had humble attempts at erotic games (which mostly looked like simple point-and-click adventure games), Japan had departments in game development companies dedicated to that. A lot of voice actresses who voiced popular anime also have voiced hentai games in their portfolio. Needless to say, the Western professional adult visual novel is still something very rare, if it even exists; the Western porn games scene is 99% indie (or 100%?). In Japan, AVNs lost the popularity they had and now are also mostly indie, but there are still several professional studios making porn games (such as Eushully) and AVNs (Giga).
Notice also how the Japanese terms are originally porn (literature) terms, not real sexual practices, while all Western terms (swinging, cuckolding, hotwifing) stem from real-life sexual practices that only later got into porn.
Btw, the site doesn't use Japanese terms for NTS; pretty sure the tag is "swinging." The problem is that this tag generalizes all kinds of Western games "involving the love interest engaging sexually with other people under the protagonist's consent" (here, I avoided using "NTS"!

). So what we need is to break one Western term (swinging) into several other Western terms.