The name Taekwondo, like the name Taesoodo, was meant to be an inclusive, umbrella term for any school that can trace its origins to the original five kwan, whether the art name was Tang Soo Do, Kwon Bup, Kong Soo Do, Soo Bahk Do, Hwa Soo Do, or anything else. That is what I think was meant by your statement above.
Incidentally though, I remember Hapkido GM JR West saying that when he was studying Hapkido in Vietnam during the middle 60's, all martial arts taught by the ROK military, including Hapkido, was considered a part of the Oh Do Kwan. It didn't matter whether you were Taekwondo or not. So in that regard, "Oh Do Kwan" was an umbrella organization for all Korean martial arts.