I had never head of that organization, either. I did do a (very brief) check, and it looks like it's less than a year old. Their web site says they're a KKW TKD affiliate, so they're expected to teach the taegeuk and yudanja poomsae. They are, of course, free to teach whatever else they want in addition to those forms. GM Chun was originally a Moo Duk Kwan practitioner, as am I. Chintae is not part of the KKW or MDK curriculum. However, there is a Chintae kata shotokan, and many of the founders of the original Kwan had shotokan training. GM Hwang Kee, the founder of the Moo Duk Kwan was one of the few who did not (there are claims that he trained to the equivalent of green belt, but he denied this). GM Hwang was primarily trained in a branch of Taichi Chuan. GM Chun could, obviously, have learned Chintae from someone and decided to add it to his curriculum, but as I said, it's not a part of the KKW or MDK curriculum.
I suspect you're right in thinking that it's the same form taught in Shotokan. I'd say it's incorrect to describe it as a TKD form, though.
I heard someone saying that GM Kouefati was thinking of making a "Moo Duk Kwan" volume 3 to complete Richard Chun's 2 volume "Moo Duk Kwan Taekwondo" series. In it he plans on including Palgwe 7, Palgwe 8, and Chintae. There is like 1 video on Youtube of some of GM Doug Cook's students doing this form. I have studied Tang Soo Do and am familiar with all the forms taught in that art such as Naihanchi becomes Chulgi, Bassai becomes Balsek, and Wang Shu becomes YunBi, but there is no equivalent of this elusive Chintae.
Yes, the TKD/TSD Chintae is a version of the Shotokan Chinte. It was taught back in the day in the Chung Do Kwan, Song Moo Kwan, Oh Do Kwan, and Jung Do Kwan. The founders of Song Moo Kwan and Chung Do Kwan learned it from Funakoshi when they studied Karate in Japan. Early Oh do Kwan days used these hyung too but were quickly replaced by tul created by Choi and Nam and since Jung Do Kwan was an offshoot of Chung Do Kwan they likely learned them from Lee or Son (respective heads of CDK pre and during the first TKD unification). There are conflicting stories on how Moo Duk Kwan founder Hwang Kee learned them. These forms are known as legacy Hyung now adays and a few dojang still teach them along side either chang hon or kukkiwon forms. The Son Duk-sung lineage of Tae kwon do Chung do kwan never made the switch to Chang Hon Tul or Kukkiwon palgwe/taegeuk poomsae due to political reasons and GM Son having a falling out with the other kwan heads and senior members of the CDK. So when GM Son emigrated to the US he taught Chung Do Kwan the same way he was Taught by its founder, Lee Won-kuk, which was basically a variation of Shotokan but with korean translations of the japanese names of the kata. A lot of schools in the New York and New England areas are of this lineage.