To clarify, Moo Duk Kwan® is an organization, not a martial art system, so when you train within a legitimate Moo Duk Kwan® school, you would be learning a martial art system instructed in accordance with Moo Duk Kwan® standards.
Before 1960 Hwang Kee’s Moo Duk Kwan® schools offered instruction in his martial art systems of Hwa Soo Do, then later tang soo do.
Beginning in 1960 Hwang Kee began introducing his new Soo Bahk Do® martial art curriculum into licensed Moo Duk Kwan® schools worldwide.
Your circumstances are analogous to those of a student who has attended a college or university (i.e., Harvard, Yale, etc.) for years and perhaps even earned a diploma from that institution. Attending a school or university and receiving a diploma for a course of study from that school does not confer rights to use the school’s name in personal or commercial activities.
For example, a university graduate might open a law firm or tutoring center using the education they acquired at the university, but they cannot use the school’s name and represent to the public that they are affiliated with that school or imply that they offer the same educational experience as that school.