Should gaming be about having fun directly or mostly as an accomplishment that leads to satisfaction? 

Is hardcore gaming more about boasting (even if just to yourself) or having fun?

I'd say it is mostly about boasting if:

After you finished a game, you go 'good, now I can sell this one and get the next one to beat'.  You let your friends know that you beat that game.  If you felt frustrated for a lot of parts of the game, but then satisfaction that you were able to beat it.  You felt on edge the entire game.

It would be about having fun if:

After you finished a game, you went ' wow that was awesome.  I'm going to play that again.'.  You tell your friends how much fun the game was, and that you made it to the end is important only to see how good the ending of the game was.  It was fun to kick back and just play the game.

While boasting is fun, is the fun the direct result of the game or indirectly thru the boasting?

The difference being running around in the game 'tag', versus running a race.  Tag is fun for fun's sake alone.  Winning the race is fun for the winner, since the runner can boast about being the best. 

Finally, does it matter where the fun comes from?  If it doesn't, why do the hardcore react so strongly against the casual gamers having fun?

I think I have both. I like to accomplish stuff in my games: I'll try to get the lowest clear time possible, I put a lot of effort in some S-ranks for RE:UC, and simply have a lot of fun being the best and perfecting my style. However, after finishing a (good) game, I'm usually like: 'Wow, great ending. Gonna play that again'. Most of the games in my collection, I beat at least twice. So I guess I'm the perfect mix then! ;)

depends on their definition of Fun.

frankly, my cat probably has more fun playing that old ball of string than I have ever done with video game. Does that answer your question? :P

You can't judge how much fun someone else is having, ever.

I'd say 31-year old women born in august in the southern hemisphere who live on the 3rd floor of a 27-story building and play games on an original GameBoy from 8:41 am to 10:09 am every tuesday and saturday morning have the most fun playing their type of games.

I don't think most hardcore gamers beat games and don't have fun while they're playing them. I know I quit if the game is boring me and when I beat a game I never sell it, I replay games quite often. I'm a pretty hardcore gamer and a lot of people are like that. Very few just play games to brag.

I'd say hardcore gamers are having more fun because they play more. If gaming was as fun to casual gamers they would play more often, unless then can't. Really, it varies with each person.

as MC Hammer once said "they don't want to hurt anyone they just want to have fun"