I think it is important to note that everyone can be a hybrid of all three.

I do it all the time, and I just wrote about it: http://news.ycombinator.com/item?id=2042017

I find that looking at the problem enables me to pick which hat to wear. Do I need to bust out some spurs and cowboy up and hack a way? Or, do I need to sit down and identify the state of the problem formally? Or, do I need to write up a specification and ship it overseas?