im confused, so is hozuki gari or not

Hozuki isnt garai but he has made a contract with tarupa just as garai did

so thoes that mean he has started a new line of shinma worriors namedafter him

I think hazuki still not the true shima worrior. He haven't inherit Garai name. He only had Tarupa, but not the true shima worrior name. Just like Kouichi inherit name as Raki and had Luche as his guardian.