I guess my own sense of badassery is similar, though I generalize that to 'anyone who stands up for something'. It could be themselves, or a loved one/goal/ideal/random rock on the side of the road that they're convinced is the most important thing in the world for some reason 
More specifically, by 'standing up for X', I mean 'do something difficult for the sake of X'. For instance, they might confront someone and face their anger/disapproval/rejection, or go on a mission that will be hard for them and cause them pain (even if the mission is 'objectively' easy/painless and it's only hard for them because they're weak).
I am pretty 'desensitized' to common acts of badassery that you always see in fiction though, like fighting a powerful enemy or stand up for something that you always see heroes stand up for. I see it all the time, so it no longer registers on my radar as a particularly badass act even though I know it would still be incredibly hard for someone in that character's position to do what they did in-universe XD
I actually consider coolness seperate though; to me, a character is 'cool' if they're subversive and stand out from the crowd without caring if anyone disapproves of this. They don't have to be subversive in an 'edgy' way; a goody-two-shoes who goes around being super sweet without caring about being called a do-gooder is cool, in my opinion Though for instance, a cool character wouldn't be badass (by my definitions here) if it's genuinely easy for them to be subversive and they choose to be that way without facing difficult consequences XD
(oh yeah, I also feel incredibly awkward talking about which of my own characters I think is badass/cool/any other positive traits that would make me feel like a fool/mary sue writer if the reader disagreed with me, so I won't answer the title question :'D)
