Your statement of what a hero does is more like a protagonist's definition.

Regarding the characteristics of a hero, which you stated, that can also be applied to a villain. A villain can have the same traits except his sense of right and wrong is twisted or flawed. Tragic villains are often born because they were the unheard or defenceless. So now they will become the voice of the unheard.

My definition of a hero is someone who puts their own well-being far below than others'. Someone who will go out of his way to save a stranger whom he has never seen before. Someone who just keeps giving and giving until there's nothing left. A selfless, kind, naive person. Superman is what I'd call a hero.

Luffy isn't one. Yes he has a sense of right and wrong and is kind. But he's selfish. He will beat anyone who stands in his way of achieving his dream. He has multiple times puts his crew in a pinch(because of stupidity) or state of financial ruin(by eating whole rations or spending all their hard earned fortune on parties).

And I've only heard him say I'm not a hero once or twice. But what he does say often is he want to be a pirate King.