Like many other kwans, Hwang Kee referred to his art of Moo Duk Kwan as Tang Soo Do (he initially called it Hwa Soo Do but quickly abandoned that). When General Choi Hong Hi coined the name Taekwondo in 1955, Hwang Kee did not choose to conform. He felt as though the name did not honor Korean history, and instead decided to change the name of his art to Soo Bahk Do in honor of Subak, an ancient Korean martial art that Hwang Kee had claimed to study (with no verification).
However, Hwang Kee was present at the meeting to establish the Korea Taekwondo Association in 1960. So why didn't he then join and finally adopt the name Taekwondo for his art?
So now, there exist three distinct schools of Moo Duk Kwan:
Moo Duk Kwan Taekwondo
Moo Duk Kwan Tang Soo Do
Moo Duk Kwan Soo Bahk Do
Essentially, all three branches were created by Hwang Kee's arrogance and instance on being the "top dog." The MDK Taekwondo broke off because they couldn't handle it anymore and wanted to unify with the rest of the kwans. Many practitioners who studied with him prior to the adoption of Soo Bahk Do continued to use the name Tang Soo Do to refer to their art, especially overseas. Those who studied in Korea and returned home often brought the name Tang Soo Do and eventually set up their own organizations once they realized that the Moo Duk Kwan in Korea would no longer support them (some even broke off to form their own kwans, such as Mi Guk Kwan Tang Soo Do which literally means the American School of Tang Soo Do).
Now, in Korea the name "Moo Duk Kwan" has since become a registered trademark owned by the Korean Soo Bahk Do Association. Do they really go after those who continue to use the name that aren't affiliated with Soo Bahk Do? I'm not sure to be honest, but the survival of Moo Duk Kwan Taekwondo and Moo Duk Kwan Tang Soo Do (and a relatively new art called Kempo Moo Duk Kwan founded by Jerry Mercado) tells me they're not overly concerned about it. In fact, due to the popularity of Tang Soo Do, many times the term is used interchangeably with Soo Bahk Do by such notable people as Hwang Hyun Chul, Kee's son and current headmaster of the mainline Moo Duk Kwan which just leads to more confusion.
So what does all of this matter? It's very easy to dismiss history as nothing more than just "politics." Surely the quality of training supersedes all of this, right? Yes, to a point. At the end of the day, no affiliation or certification is going to save your life. But keep in mind that legitimacy all comes down to what you claim. If you're simply claiming to be the best martial arts school in your area, the proof is on the mat. If you're claiming to teach authentic Tang Soo Do, then that piece of paper is essential to your credibility. And while the Moo Duk Kwan was not the only school of Tang Soo Do, since the unification of Taekwondo the majority of TSD practitioners trace their lineage to Hwang Kee. If you're not somewhere along that line, I'd start figuring out where the break happened and try to get back on course.