I think you're on to something with the idea of "if that player wants it to count". Of course, then the "best win" becomes instead "win I'm most proud of". Which may be OK -- it certainly has more inherent meaning than the current system.
The win I would like to show up on my homepage is this one:
http://www.chess.com/echess/game.html?id=8368640
You have a point, but surely some system, imperfect though it may be, can be used to adjudicate? As in, who has more material? The win counts if the player who wins was up in material or had a forced mate? Or, maybe if that player wants it to count?