No, the wording is not correct. There is no such thing as "Torath Mosheh Judaism." One can't connect the statement that way. Again, for Jews wording is key. The use of "unadulterated, genuine form of Judaism" and "versions of Judaism" are also part of the problem, English. Consider in the following way.
Now, getting to the term "Judaism" in English this term has only a modern usage and often is used to describe things that are not Torath Mosheh and are not ancient, authentic, nor autoratitive. A good example is the term Judeo-Christian. For Torath Mosheh Jews, this term "Judeo-Christian" really means Christianity and not Torath Mosheh and not Yahaduth.
- When the Torah was given by Hashem at Mount Sinai the foundation of building an Torah based Israeli society in the land of Israel was given/established.
- This included the governmental system.
- The legal system.
- The social system.
- The educational system.
- The ideological/intellectual system.
- All of the above is Torath Mosheh because both the written and oral details were given by Hashem to Mosheh ben-Amram and he taught/transmitted to all of the Israelis of that generation from all of the tribes.
- The term (יהדות) Yahaduth "Judaism" did not exist then thus the term being inserted into a time it didn't fit historically or theologically.
- Thus, anything that does not match the governmental system, legal system, social system, educational system, and ideological/intellectual system that was given at Mount Sinai by Hashem is not Torath Mosheh.
- In order for something to be proven to be Torath Mosheh it would have to be proven to be ancient, authentic, and authorative. I.e. you should be able to trace the concept, practice, etc. throughout the generations and find it common in the most ancient Jewish communities.
- If something does not match the above it is not Torath Mosheh and thus Torath Mosheh reject it.
In all ancient sources where the term (יהדות) "Yahaduth" is found it does not mean what the English term Judaism is used in the western world to mean. Thus, now a days you have Christian Jews who claim to follow "Messianic Judaism" BUT if you go back prior to the 1960's no Christian Jew ever used the term (יהדות) to describe what they were doing. They originally called themselves "Hebrew Christians." At no point in their history did they describe what they were doing as Torath Mosheh.
Well, from now on with you, I will only use the term Torath Mosheh Jews. Also, there sure would be a lot of people who would disagree with/would not like you saying that their version of being Jewish isn't really valid and that only Torath Mosheh Jews are.
As an experiment do the following. Take anything that has the term "Judaism" attached to it in English and see if that thing was in practice or accepted among Yemenite Jews, Mizrahi Jews, Maghrebi Jews, Asian Jews, etc. Further, look at the origin of said thing and if you find it only started in particular modern year and had no predecesor in earlier mentioned ancient Jewish communities you will know that said thing is not Torath Mosheh.
That sure is quite an undertaking. However, it would be helpful to me if you could provide a list of these Judaic groups. Or I could look at the information in the Wikipedia article that you seemed to have dismissed.
At that point, you state, "This is not a matter of it being unadulterated, genuine Torath Mosheh. Said thing is not Torath Mosheh without arguement and those who hold by said thing, on thier own, never claimed it to be Torath Mosheh."
Well, please don't be offended by this, but what you're explaining reminds me of fundamentalist Christians groups such Jehovah's Witnesses, Seventh Day Adventists and others who claim that other forms of Christianity are cult-like groups and that only they are the true Christian faiths that can be traced back to what the Bible actually says and back to apostolic times.