Several threads on here have popped up with people arguing, like me, that story isn't important to games, while others have come out and said that stories rule all, and mature people care about stories in games.  Well, I figured I would look at what I see as a spectrum of play and forms of entertainment.  I see the spectrum going as follows (open-endedness vs narrative driven, or structureness):

1. Toys.  These are things you play with with no point in mind.  You have a degree of control over them, and make them do what you want.  These aren't too frequent in videogames although something like Noby Noby Boy would likely fit here.  If a game comes with a development language also, to create content in a very open way, it could be considered part of this also, like something like Kodu.  I would put Sony Home here, as is Second Life.

2. Puzzles/classic arcade games.  These are like toys, but are goal driven, in that you need to get the toy to a certain state.  Tetris can be seen as fitting into this.  Classic arcade games like Asteroids and Space Invaders would fit into this here also.  Rather than solve problems, you continue to play at them.

3. Sandbox games.  These are open play areas, aka a sandbox, but will have puzzles or other challenges in them.  Sandbox can fit Grand Theft Auto or Crackdown or inFamous, or they can be like Civilization or Simcity.  The former ends up likely having more story put on top, while the later gravitates towards the toys/puzzle category.   I put this after puzzles/classic arcade because I am think GTA, which will have an overarching story involved with them (act as a way to string together other challenges).  In games like Simcity, I would say they go before puzzles/classic arcade, when they are more like toys.  Western RPGs can fit here also, depending on the open-endedness.  I would say LittleBigPlanet may fit here actually, or anything with a level editor for it.  World of Warcraft likely fits here also, or another Online RPG with a world that dynamically changes.

4. Sports and boardgames.  These are competitive ventures, where the object is to use the play area to defeat an opponent.  The approach is fairly wide open, but the goals are set.  Not as open as sandbox games, but still more open than other formats of play.  Chess would fit here, as would RTS, or sports games, or games like Settlers of Catan.  Fighting games definitely belong here to, as do driving games, and also multiplayer FPS.

5. Post-arcade console titles, with endings and levels.  Sonic, Mario, Zelda, and others that we know, fit here.  The games have endings, and don't go on indefinitely.  This is the classic videogame as we know it.

6. JRPGs and single-player FPS titles.  These are fairly linear in nature, but some can be less linear for players on a whole.

7. Adventure games and interactive fiction.  These are very strict in how they are laid out and linear in nature.  Text adventures fit into this.  And on the more strict end, a game like Heavy Rain would fit into this.  For individuals heavily gravitating toward story and character, this is a place they would consider this the best form of play.  I would also count Uncharted in here also (single player).

I write this to show there is a WIDE spectrum, based on the degree of player control over the environment, or the environment dictates how they play.  I am sure people may want to end up moving (swapping arcade and sandbox, for example) or consolidating what I wrote.  Feel free to.  This is not an end all and be all what I wrote, but a starting point to discuss things.

I would say though, can we live and let live a little and respect people prefer to play on one end of the spectrum, while others want things on the other end.