All of this is your head canon. Sword skill and Hakimanship are just as different as sword skill and DF users.
The entire template and those sections you're trying to put it in is your headcanon. And your head canon that King/Zoro have already debunked.
This is Oda (through King) personally debunking you.
Those "4 major powers" only exist in your head. Not in the manga. In the manga, there are any specific categories for that. Except for Swordsmen. Which Oda consistently makes clear. Swordsmen exceed in "sword skill". And only those are declare themselves as swordsmen are considered as such. Confirmed by Zoro himself. Swordsmen do not = "those that use sword".
Irrelevant. Kaku calls himself a swordsman (if he did, so be it). Shanks/Roger haven't and therefore they ain't swordsmen.
Nah wrong, he specifically stated that he would do anything to win. So in other words, it was literally Zoro confirming that him ripping his throat out wouldn't make him a swordsmen. Whatever the facts are clear according to Zoro;
1. You ain't a swordsmen if you don't consider yourself one
2. Having a sword doesn't make you a swordsman
3. Having a duel with a swordsman doesn't make you a swordsman