As I recall, Haedong translates to 'East Sea.' Not completely sure on that.
Haidong and haedong gumdo are essentially the same thing, but there are two different HDGD federations, one using haedong, one using haidong. Gumdo and kumdo use the same characters, but kumdo spelled with the 'k' generally refers to kendo in Korea, while gumdo usually refers to HDGD or Shim Gumdo (on which HDGD is partially based).
Haedong gumdo is more akin to kenjutsu and iaito, while Daehan kumdo can encompas these, but does not always. Daehan kumdo is essentially Korean Kendo. It is very focused around kyorugi (sparring) with the jukdo (shinai), wearing hogo (protective gear, called bogu in a kendo school) and is a form of fencing. Generally, kumdo has forms and some muk do (bokken) techniques. Some kendo/kumdo schools incorporate a full curriculum of kenjutsu and iaito (our school does this, though you have to be around for quite a while, as traditionally, iaito and kenjutsu are studied as separate martial arts). Some kendo/kumdo schools are nearly entirely tournament/sparring oriented.
Haedong Gumdo is a relatively recent martial art (1970's I believe) with roots in Shim Gumdo (another relatively recent sword art, 50's-60's I believe) and according to the founders, ancient Korean sword arts. The style revolves around the katana-like daedo, and from what I have seen, has more in common with kenjutsu than Korean sword styles (If I'm not mistaken, Korean swords generally hung from the belt with the curve of the blade facing down, like a sabre, while in HDGD, they wear the sword through the belt with curve facing upwards like a katana), and is clearly influenced by it in my opinion. Not really a problem, as there was a lot of sharing of techniques and such between Japan, Korea and China anyway.
HDGD claims a two to four thousand year old lineage with lings to the hwarang and the samurang. While I find any such link tenuous at best, a lot of kumdo and taekwondo schools do the same sort of thing, I can't single them out for criticism in that area. To my knowledge, there isn't enough historical information on any of the ancient Korean sword to put together a comprehensive and accurate curriculum. I could be wrong, but that is what I understand.
HDGD is a legitamate, though recent sword art and it is a Korean art, which is nice for those who have interest in easter sword work but want somthing other than kendo/kenjutsu or Chinese styles. It is focused on use of the sword in battle, rather than on the duel, which is what kendo/kumdo is more focused on, much like western fencing.
Hope that helpes. If anyone here is an HDGD practitioner with information, please jump in; my info is all second hand. Ninjamom is tremendously knowledgeable and is an HDGD practitioner, and I'm sure that there are some others here.
Daniel