Good points.
You have heard of Rabbi Bar-Hayim right? From Machon Shilo. I believe he proposes calling "Orthodox Judaism" Halahic Judaism (at least according to the WikiPedia page about him).
Since the term "Orthodox Judaism" came about in response to the other movements such as reform. Before that Judaism was just Judaism. I happen to agree with him, though it would be virtually impossible to change anything now since people group Judaism as "reform" "masorti/conservative" and "Orthodox". I happen to feel... "reform" and "conservatives" have their own ideas, they can call themselves whatever they want, but Judaism...the same one Moshe Rabbeinu brought us, that King David followed that the Prophets followed...THAT is real Judaism. My friend actually told me he went to a religious forum at his college and there were three men speaking on behalf of Judaism. One reform, one conservative and one Haredi. The reform man said "I am rabbi so and so and I am a Reform Jew" the other man said "I am rabbi so and so and I am a Conservative Jew" lastly, the religious rabbi said (with a pleasant smile under his black hat) "I am rabbi so and so, and I am a Jew".
Your point just reminded me of that because I had recently heard about calling Judaism "Halahic Judaism" and then you mentioned people who claim their Orthodoxy to the Hattam Sofer.
I'm sorry I've gone off topic.