In Japan, American shows fall under {海外|かいがい}ドラマ (foreign drama) whereas domestic shows are just plainドラマ (drama). I'd hazard a guess that the English terms C/J/K-drama are reborrowing from the broader Japanese or Korean use of "drama" to describe shows, since I don't know about you but I would rarely say I'm "watching a drama" in English. Those terms predate the mainstream Western awareness of East Asian dramas, so it makes sense to me that enthusiast groups would draw from native terminology in the same way that we use the Japanese demographic categories shonen/shojo/seinen/josei for manga and anime.

Also, it's not limited to Asia--see Britpop, NWOBHM, Eurodance, Eurobeat. The main reason you don't hear more about, idk, F-pop/drama is that there's zero market penetration for French music and shows in English-speaking communities. On the other hand, I hear plenty about British TV shows and when I'm talking about them I usually identify them as such.