Board Games: Story or Contest?

During a recent game night with the board game Descent: Journeys in the Dark, I realized an interesting dichotomy between why my friends play board games.  Some friends want to play the games to have an adventure/live the story, while others want to just beat their friends in a competition.  (Well actually, it's more of a trichotomy because one of my friends comes mostly to socialize and plays the games because that's what we're all doing).  After realizing this, I thought about how I play and I think that I switch between the two depending on the game.  To me, Descent is D&D Lite, so I like to play that more as an adventure.  Other games, I kind of just want to win despite the interesting story of the game.

Before I go on, I have to define a few things.  Theme is not the same as the story.  A game's theme is the setting and style.  The who are you and what are you doing of the game.  Every single game has this.  The story is the why of the game.  Why are you trying to achieve your objective?  Not every game has this.  Apples to Apples theme is comparison and modern day pop culture.  Settlers of Catan's setting is the island of Catan.  If it has any story it's that you are competing to settle the island better than everyone else because I guess something magical happens once you settle it enough?  Typically, the games with stories are the ones where you have characters and are playing a role.  The Mansion of Madness, the Betrayal at House on the Hill, the aforementioned Descent and Adventurers games, Tales of the Arabian Nights, etc.

So, obviously if a game has no story or a weak story, people are playing to win (unless they have a fantasy world in their head even more elaborate than my own...).  But what I've discovered personally is that when I play for a story, the game is more fulfilling than just playing to win (unless I've been on a really long losing streak...).  Some of my favorite game nights are ones where a story came out of it.  Like the time my friends were playing Tales of the Arabian Nights and my friends spent the entire game in prison, ensorcelled, insane, and enchanted, finally escapes by grabbing on to a rope of an evil visier's magic carpet, and then falls to his death because he can't hang on.  (That was not such a great night for him and he won't play the game anymore, but I still find it funny.)  I think part of the reason why playing for story is more fulfilling is that I tend to play nicer.  I'll hint about strategies other players could employ against me because it will make things more interesting.  I might try riskier strategies because if it succeeds, it's going to be an awesome story and if it fails I can go down in a blaze of glory.  Also, when I play for story and don't play to win, there's less chance of me pouting if I reach the point in a game where it's impossible to win and I have to keep playing (being trapped and unable to build in Settlers is so depressing).

What I'm kind of curious about, is when designing a game do the designers have a story in mind?  I know for  every game I've come up with there is always a definite story in mind for the players to experience.  The game always takes precedent over the story when it comes time to design it or make choices, but it's always there in the back of my mind.  I guess that's what the rule book is for...