I was thinking about Red a while ago and how he's lacking all kinds of characterization in the games. While you could say that he's the player's avatar in the first games, that doesn't aply in the sequels anymore, as you're now taking the role of Gold. Red being on top of the most dangerous area of all Jotho and Kanto combined, standing in the sun, having the stronges team and more importantly: refusing to talk to you, or anybody, kind of lead me to one possibility: Red's a jerk
While in gen 1 Green/Blue seems to be the jerkish rival who always picks on Red, we never actually know, why he's behaving like that. It is possible, that Red himself acted the same way, resluting in both of them trying to best the other one.
One reason why Oak thought of Red being that super nice and polite neighbor kid could just be, because he always played the nice guy in front of adults.
So, why is he standing on top of Mt Silver? Because he considers himself to be superior to everybody else, which he shows by climbing on top of the assumably highest mountain, where he looks down upon all the other people, who have to look up to him. Beating him, just triggers his rivalry behavior, resulting in him training more and returning when he feels capable of taking you on.
Or it could just be a reference to what happens to the player when he gets beaten: he returns to the Poke Center.