I'm looking for a list of game modes, but first, what is actually a game mode?
But first, why are you going all philosophical about it? Are you trying to figure out a list of game modes for your game?
Wikipedia is good but perhaps it's also of some help to think about how game modes could be planned and added:
1) Single player, Splitscreen Multiplayer, Network Multiplayer are often decided in pre-production because they hugely affect the budget and needed skills and resources. Multiplayer and Singleplayer traditionally have distinguished gameplay with single player focusing on story and multiplayer focusing on interaction between players, whether it is co-op, versus, team, deathmatch, capture the flag...
2) Usually during the development process the authors and game testers might find that some section of the game has unique gameplay and want to make that part easily replayable. They could make it a minigame inside the game, like Snowboarding and Motorcycling in FFVII but they might as well make it an additional game mode that unlocks after you've beaten the game.
3) Developers can add additional game modes that highlight one aspect of the game. Batman: Arkham Asylum is good example on this with it's "Challenge Mode" covering Stealth, Fighting and Predator each of which are distinguished game modes with different gameplay and rules. Mercenaries mode available in some RE games highlight the intense action part of the game, leaving out the puzzle and story parts.
4) A lot of random things like Time Attack modes in racing games or extra-hard difficulty levels are an example of re-iterating through your game content with different ruleset and gameplay. Some of these aren't even that meaningful or hard to achieve like Reverse Tracks / Double Speed mode but they could be viewed as adding gameplay content. Because you're using the same content they often don't require huge amount of planning or resources so you can decide on these depending on your genre and as you develop your game. You can even add them post-release as DLC or patches if you still have motivation for it.