Somewhat old post but...
Apart from the contradiction there, you already -have- hats for these purposes, or these purposes would have come up in your original request. You're looking for a hat for which -neither of these conditions apply-, or else they would have entered into your specification.
That's not necessarily true. These conditions may have been necessary but not sufficient conditions.
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?