As I am sure is the case with many people here, my desire to bike commute results from more than one of the reasons you have listed. But I ended up picking the "I like it when people think I'm brave/strong/weird" category, though not primarily because I care what other people think of me. When I reflect on my decision to bike commute, that answer seemed to fit best with the real reason.

I like to challenge myself both physically and logistically, and bike commuting in my situation definitely does that. I also tend to operate a little outside of the mainstream, though I suppose most bike commuters do. And I like being outside (physically outside, not in a car). I guess you could say that I like it when "I" see myself as brave/strong/weird. Because I would do it even if no one ever saw me or knew that I did it.