I have mentioned before that games convey moral principles, sometimes through a deliberate decision by the game designer. Even though I'm a lurker's lurker at The Forge, I'm aware of the taxonomy contained in the GNS model. It should be of no surprise that one key mind behind the ongoing development of the GNS theory is a biologist (with a penchant for bat genitals and the sounds that rats make after sex, it seems). One of the building blocks of biology is the taxonomical structure (thank you, Linnaeus) which breaks down all life into a hierarchy, and which allows for unique naming to take place. Thus, the GNS is a taxonomy.
But as I've said, games convey moral principles. They also convey philosophical perspectives. This, therefore, is not the biologist's taxonomy of gaming types; it is the philosophical taxonomy of gaming types. It should be of no surprise that the mind behind this development studies theology and philosophy (with a penchant for bat food - the divine mango).
In which the game is about creating a story after the fact, looking back on the actions and decision points to create a story. This is probably the most common of the story game types. However, the mechanisms of the game probably prevent this from being a pure existential game, since those mechanisms incline the character towards certain decisions - that's incline, not force. The player can still choose to have the character go against the option with the most likely success.
In which the game is about decision points that favour neither side of a conflict, and the story still moves forward. This is the clash of two opposing motives creating a new situation that contains elements of both, but which is not actually either. Typically this is seen in games that allow for two characters to oppose each other, fail, and still the plot moves forward. Make failure interesting through the dialectical approach.
In which the game is about the characters assuming and perfecting a role in the cosmos, with no necessary interest in the big picture. Any game which is just about levelling up the characters is really about a character not simply accepting their fated raison d'etre, but embracing it and rushing towards it with all enthusiasm and vigour. Fatalism should not be confused with fate; it is more to do with asking, "What was I meant to do in this life?"
In which the game follows an established narrative form, with structured events and decision points. Any game that has an endgame mechanism is like this, in that the overall narrative has an end. The form of the story always results in an encounter with The Big Bad (or similar), but there is no guarantee that the protagonists will triumph. Any game that will always result in the characters dying, or going insane, or falling asleep, etc., is a formal game.
In which the game reactively follows the trumphs and tribulations of the characters; bending the will of non-player protagonists and the acts of objective material around the existential history of the protagonists. This is typically found in games in which the player can bank points from character suffering to later spend on changing the future (usually for character favour). The character is the centre of the universe (whether actually stated in the game or not) because of this mechanism of balance, and therefore the character may as well be the only character in the universe.
This is far from a complete list. What philosophical forms have you seen in games?