I have been silent on this topic for far too long. So I will give you my 2 cents which probably isn't even worth that.
My instructor was in Korea when Hwang Kee opened the Joong Ang Dojang (Central MDK Gym). He has watched it grow in Korea and helped to spread it in Korea as well as here in the USA. With that said, I will give you what I have told my students which was shared with me by my instructor.
Originally Hwang Kee called his school Hwa Soo Do from 1945-1949 or so. But no one understood what that meant, and he was unable to get/retain students using this term. So Hwang Kee changed the name of the school to Tang Soo Do Moo Duk Kwan (1949 or so thru 1960/1961). He later changed it to Soo Bahk Do Moo Duk Kwan in the either 1960 or 1961. Hwang Kee created the Korean Soo Bahk Do Association (around 1960 or 1961) of which my instructor claims to be a member of to this day. When the South Korean Government pushed for a unification of the Kwans, they created the Korean Tae Soo Do Association. And then eventually changed the name to the Korean Tae Kwon Do Association. It was around 1965 that two senior Moo Duk Kwan students eventually joined the Tae Kwon Do movement. Keep in mind that these two students had spent three or four years attempting to convince Hwang Kee to join the Tae Kwon Do movement without success. Hong Chong Soo (sometimes Hong Jong Soo), Dan Bon 10, and Lee Kang Eek, Dan Bon between 13 & 25, are the two senior students that joined the Tae Kwon Do movement. Others followed these two over to Tae Kwon Do. And some remained with Hwang Kee and Soo Bahk Do. This was the first major fragmentation of the Moo Duk Kwan, and the birth of Tae Kwon Do Moo Duk Kwan.
Keep in mind that Korean instructors from the Moo Duk Kwan came to the US between 1960 and 1970. These instructors used the term Tang Soo Do Moo Duk Kwan. In 1975-1976 Hwang Kee dispatched his son, Hwang Hyun Chul (Dan Bon 509), to go to th USA to take over the creation of the US Tang Soo Do Moo Duk Kwan Federation. Due to various reasons there was an exodus of senior level Moo Duk Kwan practitioners between 1976 and 1993. There is a lot of speculation as to why this occurred. Some say they were forced out. Some say they were unhappy with the direction and chose to leave. Regardless, I would argue that this was the next major fragmentation of the Moo Duk Kwan.
During the years between 1975 and 1993 the US Federation was using the term US Tang Soo Do Moo Duk Kwan Federation. In 1993 they changed the name in the USA to the US Soo Bahk Do Moo Duk Kwan Federation. This is because they were unable to enforce the copyright of the term, "Tang Soo Do Moo Duk Kwan." The US Tang Soo Do Moo Duk Kwan Federation had sued many senior Moo Duk Kwan practitioners that were using the term Moo Duk Kwan. They had varied and limited success with the enforcement of the term Moo Duk Kwan. Some practitioners changed their belts to be more Japanese such as Shin Jae Chul (Dan Bon 698) who founded the World Tang Soo Do Association. Others changed the movements of their basics and forms to avoid a lawsuit such as Kim Chun Sik (Dan Bon 2457) who founded the International Tang Soo Do Federation. Others such as Kim Jae Joon (Dan Bon 38) were relatively successful in fighting the copyright issue. And others were pretty much untouchable such as Lee Kang Uk in Britain with his U. K. Tang Soo (Soo Bahk) Do Federation - Moo Duk Kwan. This was due to the fact that the copyright was filed in the USA. Therefore not enforceable in the U.K or Korea for that matter.
In conclusion there are really three major branches of the Moo Duk Kwan. The Soo Bahk Do folks who are under Hwang Hyun Chul at present. The Tae Kwon Do folks who are aligned with the Kukkiwon and quite probably the World Taekwondo Federation. And then the Tang Soo Do folks who are holding onto the original forms (Pyung Ahns) taught.
Now in response to the phrase "there is only one Moo Duk Kwan", this is a ploy to discredit those that are using the term Moo Duk Kwan. The US Soo Bahk Do Moo Duk Kwan Federation is known for using different tactics from legal threats to pure outright trying to convince the consumer that these Tang Soo Do Pioneers are frauds. In fact, the US Soo Bahk Do Federation even purchased "KarateFraud.com" in an effort to bully those that use the term Moo Duk Kwan.
The reality is this...Hwang Kee wrote in the book Soo Bahk Do Dae Gahm (Korean Version), published 1970, and translated into English as Tang Soo Do (Soo Bahk Do), published in 1978 that he believes in Ryu Pah. Essentially, a river flows down the mountain and divides on its own. If this is the case, then according to Hwang Kee all of these flavors, if you will, (Tang Soo Do, Soo Bahk Do, and Tae Kwon Do) are Moo Duk Kwan. I look at Hwang Kee's actions and he accepted several members from other Kwans as Black Belts. They were given Dan Bon numbers even though they received there black belts at one of the other Kwans. I feel this is how Hwang Kee views things.
Sadly, I am not sure why the US Soo Bahk Do Moo Duk Kwan Federation has done and continue to do what it does to its Seniors. I have spoke with many senior Moo Duk Kwan practitioners about this including Hwang Hyun Chul himself a few weeks ago. It is my understanding from Hwang Hyun Chul that he is attempting to mend bridges by creating and promoting the Moo Duk Kwan Heritage program. I am not sure how this will play out. So far, my instructor doesn't seem to have any interest in this program much less mending bridges at this juncture in his life. I, myself, do not care for the . Hwang Kee was a person that promoted character development and world peace above all else. And I feel that this phrase, "there is only one Moo Duk Kwan", is not what Hwang Kee would have wanted as his legacy.
As an interesting side note, the Moo Duk Hae in Korea is a group of Moo Duk Kwan practitioners that get together a couple to three times a year socially. The only requirement is that your lineage can be traced to the Moo Duk Kwan. This group does not care if you are Tae Kwon Do, Tang Soo Do or Soo Bahk Do. In fact, members of all three groups can be seen at these social gatherings. I believe this is in the spirit of what Hwang Kee would have wanted. And it is my hope that this is the direction that Hwang Hyun Chul will go with the Moo Duk Kwan Heritage program. Only time will tell. Anyway, I hope this information is helpful and useful.
Please bear in mind that this is my viewpoint and not necessarily that of my instructor. Any errors are strictly mine and mine alone.