The term Moo Duk Kwan Taekwondo can have several meanings. Moo Duk Kwan was one of the original Nine Kwans of Korean martial arts that arose after World War II.
The term may refer to the style of Traditional Taekwondo practiced at Moo Duk Kwan during the 1950s and 1960s.
The style of martial arts practiced by Moo Duk Kwan eventually went on to be called Tang Soo Do. In the U.S., the term Moo Duk Kwan Taekwondo was sometimes used to refer to Tang Soo Do.
Eventually one faction of Moo Duk Kwan (like the other Nine Kwans) went on to embrace Kukkiwon/WTF-style taekwondo. The term can refer to Kukkiwon/WTF-style taekwondo as practiced today at Moo Duk Kwan.