They can be necessary but not sufficient conditions for buying a hat. The issue is that even once those conditions are met, those two reasons aren't merely reasons for buying a hat, but important specifications for the hat you are going to buy. If your goal is to prevent heat stroke, a hat that keeps you warm in the winter is counterproductive.
(I have owned quite a few hats, and used to wear one most of the time in public - in my college years a bowler, later a fedora, by which I mean a fedora and not a trilby. I donated most of them after realizing the effect they were having on my mood. At this point I own two hats; one is a black oilcloth hat for rain and costumes, and the other one is a fabric hat for sun.)
For years, I've taken being a classic nerd, geek, and hacker as a point of pride - eg, consciously trying to judge people by the code they produce, or whatever else they write, as opposed to judging them on their appearance, to the point that I prefer /not/ to know what my favourite authors look like. I've tried to make what strengths I can out of the resulting weaknesses, such as reducing decision fatigue by keeping a single hair-style for many years, wearing whichever t-shirt is on top of the clean shirt pile, and so on.
I'm no longer satisfied with this. I want to become stronger.
Last month, I bought a dozen button-up, collared shirts... and have noticed slight, but consistent changes in my workflow when I wear them. I want to leverage whatever other clothing-based self-improvements are within my budget.
For some years, I've worn a floppy boonie hat to shade my delicate eyes from the burning rays of the sun. (I've even been seen wearing it with a photographer's vest instead of a daypack while tromping around my hometown.) I'm thinking of trying out the 'Crasche' safety inserts mentioned in the recent Open Thread while hiking far from medical help, which would require a baseball-like cap with a sweatband. Given my proclivity for taking something that works and sticking with it for years, I might be wearing that cap for a very long time.
Thus, a multi-layered question: Which hat should I buy? Which factors should I take into account... and which shouldn't I? Are there any subreddits, forums, or other online discussion groups whose members would be willing to take this question seriously and with only a minimum of mockery?