Why Categorizing Games is Difficult
A "real game" is made from a combination of multiple game elements. A real game might be a 10k marathon skeet shoot with a round of rock-paper-scissors when it starts. The basic game elements can be arranged within a real game in infinite ways.
So when it comes to categories, real games are messy.
Games are Not the Sum of Their Parts
You can't categorize games by adding together the game elements from which they are comprised. The overall play and goal of a game are not a summation of the play and goals of the sub games. No particular part of a game matters; only the game as a whole.
For example, you can't call a ten mile run the simple equivalent of ten one mile runs. The final mile is significantly different from the first, despite being an identical game element on paper. The first mile requires you to run after having been at rest, while the final one requires you to run after having immediately run the previous parts.
You don't play a game the same way that you play each part of the game. You may win nine parts out of a game's ten and lose, or you may win one part out of ten and win. If playing against multiple opponents, you could lose every part and still win.
Games are not Goals
"Playing games" is not the same thing as "winning games". The win condition of a game is added as a guide to players to make certain decisions and to provide a stopping point. Games are nearly always about playing and not about winning.
I play many games where I don't even know what the win conditions are. Apples to Apples has some win condition regarding the number of green apples you've collected; I simply enjoy making the associations. I could plan strategy and psychology, carefully assess how the other players choose the winning cards, and maybe play better. Maybe I would win more. Not only do I not care, it would ruin the point of the game. I still hope my card is picked, but I refuse to do anything more to ensure that it is.
I play Anagrams very competitively, but I never check at the end of the game if I've won. I don't even know what winning is in Anagrams. Most words? Most letters? Longest words?
Playing a game doesn't necessarily utilize the same skills as playing to win. Winning Scrabble requires you to balance points, positions, and tiles. Playing Scrabble might involve finding relevant words or cool patterns.
For every game played to win, thousands are played for the experience, where the winning is only another rule to guide your decisions. It doesn't mean that some semblance of playing toward to win doesn't happen; it's just not the focus.
Games are not Play
Games are played very differently at different times. A game can be played seriously or not, cooperatively or competitively, and with more or less interaction. The rules of the game can encourage certain type of play, but cannot force it.
A foot race can have deep interactive psychological elements while boxing can be played almost entirely in a Zen state of mind.
Is Settlers of Catan interactive? Is it competitive? Is it goal directed? Only if the players make it so. The only thing you can absolutely say is that players roll dice, pick cards, and trade those cards in for tokens to place on the board.
There are games whose interactive and cooperative aspects are designed to change from one play to the next. For instance, Cosmic Encounter is a game that can have a lot or a little interactivity depending on the number of players and the cards picked each game.
The type of play is affected by the length allotted to play the game, the number of players (which is not the same thing as the number of player positions), the ages and personality of the players, and the rewards for winning or losing. Because of this, categorization that tries to sort games into play types is problematic.
The Immutable Aspects of Games
Instead of categorizing by play, it seems reasonable to categorize games based on the immutable aspects of the game, which are those things about the game that are determined before play begins. They are independent of the play.
Physicality
If a game has any physical performance component at any point during the game, it is categorically different than a purely mental game. Mental only games can be played by proxy via telephone, email, etc. Physical performance games always require the player himself or herself.
Categorizing within physical games gets messy again. The physical requirements can change during the game. You may be required to do one second of manual dexterity followed by an hour of running, or play a game of Chess and then throw a punch.
Some thinkers declare all games with any physical aspect a "sport" which seems a bit of a stretch to me. By that definition, nearly every video game is a sport.
Equipment
There are five major types of equipment: Bodies (a physical presence), abilities (throw the ball), spaces (an appropriate location), objects of a specific nature (a lawn dart, a network connection), objects of a general kind (a play surface).
In addition, two related categories of games are their portability and accessibility.
Theme
Many games are abstract. A sports game is typically abstract, while a board game about sports is not. Like literature, music, and movies, games often overlap multiple genres. There are fantasy sports games and games that cover multiple historical periods.
It is questionable as to whether a theme can really be used to categorize a game. If you add a television theme to the game of Checkers, is it a different game? This is unclear. You could call it a different game, or you could call it the same game with a thematic overlay.
Player Positions
Player positions refers generally to the number of teams in play. If categorizing using player positions, then a three-player game is a different beast from a four-player game, even when using the same components and rules.
While considering player positions, you can also categorize games by their position balance. Some games present equal opportunities for each player. Others use different but balanced forces. Others are unbalanced by design; for instance, one player may have a starting advantage or disadvantage.
Turns
Games with rigid, solitary turns feel different than ones with simultaneous play. Some games have simultaneous play but require some event coordination. For instance, after a goal is scored, all players must stop play until a signal is sounded. Other games have full or limited play opportunities available to all players regardless of whose turn it is.
Complexity
Complexity can refer to the rules, to the depth of play (lookahead), or to the breadth of play (distinct multiple options).
Activity Categories
Absent from this list of categories is what you are asked to do in the game: capture, hold, shoot, race, auction, and so on. That's because these types of distinctions are messy and ill-defined.
Furthermore, they depend on the players understanding the game's theme. A rule set that tells you to turn over a card and then pick up another card is not telling you to capture, hold, shoot, race, or auction; it is the theme which is doing so.
Further Reading
The Definition of "Game"
Game Mechanics Don't Really Exist
http://people.stfx.ca/dvossen/3-CV/AVANTEPaper2004.pdf (PDF) - On physical (sport) vs non-physical, competitive vs non-competitive, and interactive (must be competitive) vs non-interactive.
http://en.wikipedia.org/wiki/Game_classification
http://en.wikipedia.org/wiki/Game_theory#Types_of_games
http://www.gameclassification.com/
http://www.robinlionheart.com/gamedev/genres.xhtml
http://www.gamethink.net/For-a-new-classification-of-game.html
http://www.gamasutra.com/features/20031003/lindley_01.shtml - On classifying video games.
http://www.pagat.com/class/ - On classifying card games.
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