But if you are inside the bubble of warlords then you are a pirate, that's the thing. It's a group made of pirates whose piracy is sanctioned, not a group of independant sailors.
"Looting" as definition of being a pirate seems quite simplistic in this particular context too, considering that in One Piece pirates may not even loot but simply live their lifes as free men, establish a huge empire that works like a country, and so on. I doubt there's such "being a pirate" concept in this story, and if it is, it's probably more related to doing whatever you want because a pirate is free.
I see no reason to overcomplicate this. The Shichibukai is described as a group of pirates minimum three times in which Mihawk is included. No professional bounty hunters like Jean Ango, no independent players like arguably Kuzan, only pirates.