Games may be classified and sub-classified according to many different criteria. Each scheme has its own advantages and disadvantages. The categories on the main game page is rather a hodge-podge, and might benefit from some rationalization. However, before undertaking to refactor the game pages, please give thought to all of the following distinctions.
What sort of challenge / skill is involved (e.g. abstract calculation, anagramming, luck, bluffing, verbalizing, coordination, speed, etc.)?
Leads to the ("Folk Model") theory of 4 categories: games of skill, games of chance, games of strategy, games of status propagated by Anderson/Moore and Brian Sutton-Smith.
This scheme is probably most natural, and quite neatly separates billiards from chess from Tomb Raider. The main disadvantage is that too many games fall under more than one head. For example Scrabble relies a great deal on word knowledge and anagramming, but also has significant strategic aspects.
What equipment is used to play the game (e.g. a computer, a board, cards, tiles, dice, etc.)?
This categorization is also very natural and common, but sometimes problematic. For example, Balderdash is a commercial board game, whereas Fictionary is almost identical but uses no board.
Something is odd with any scheme which forces such similar games to be listed under completely different headings.
Other distinctions are less important, and apply more or less well to different major headings. For example, example, the difference between team and individual sports is fundamental, whereas team board games are so rare as to hardly merit a category. The remaining distinctions apply mostly to non-physical games.
How many players does the game accommodate? The most important division is between two-player and multiplayer games, because nearly all multiplayer games involve negotiation or coalition-building to some degree. Among multiplayer games it is also important (particularly to whomever is organizing the party) what range in the number of players can be accommodated. One disadvantage of this distinction is that a few games such as Titan are equally good two-player or multiplayer.
To what extent to which chance is a factor? Games run the gamut from having no chance whatsoever (checkers, Pente) to being entirely determined by chance (roulette, Chutes and Ladders).
How deep is the strategy? Some games (bridge, Go) can be studied for years without exhausting what there is to learn, whereas others (Three Men's Morris) can be mastered relatively easily.
How easy is it to learn the rules of the game? Chess and Go are often compared for their depth and abstraction, but chess has considerably more difficult rules. This consideration is particularly important for family games, where ideally children should be able to play along easily, without making the game so simple it holds no interest for adults.
Is the game relatively abstract or does it attempt to simulate some aspect of reality (e.g. stock market, war scenarios)? For some simulation games, the realism is more important than all other factors, whereas some games (Set) are so abstract that the names and shapes of all the pieces could change without affecting playability. However, most games lie somewhere in between, with a balance between abstraction and simulation.
Are players eliminated as the game progresses, or can everyone play along until the end? This is most important socially, as a host may wonder how to entertain guests who have been knocked out of the main event.
What is the objective of the game? This is most useful as a sub-subheading, because different types of games tend to have different types of objectives. For example card games have natural categories of trick-taking and shedding games, which don't apply to board games, whereas board games have categories of capture, racing, and immobilization which don't apply to card games.