A game mode is a distinct configuration that varies gameplay and affects how other game mechanics behave. A game with several modes will present different settings in each one, changing how a particular element of the game is played. One of the most common examples of game mode is the single player versus multiplayer choice in video games, where multiplayer can further be cooperative or competitive.
Changing modes while the game is ongoing can be used as a means to increase difficulty and provide additional challenge, or as a reward for player success. Power-ups are modes that last for a few moments or that change only one or a few game rules; for example power pellets in Pac-Man give the temporary ability to eat the enemies for a few seconds.
Other examples include the availability of a sandbox mode without predefined goals or progression. The division of game content in stages or chapters, where each stage expands the rules that a player can use with respect to the previous stage, increases game complexity and variety. If the game advances through these stages by moving through different areas, these areas are called levels or maps; if the character unlocks new abilities through activities or rewards, they receive a currency called experience points. These points can be used to upgrade or augment various pre-determined abilities.
A game mode may restrict or change the behavior of the available tools ( e.g. play with limited/unlimited ammo, new weapons, obstacles or enemies, a timer, etc.), establish different rules and game mechanics (e.g. altered gravity; win at first touch in a fighting game; play with some cards face-up in a poker game) or even change the overall game goals (following a campaign or story vs. playing a limited deathmatch or capture the flag set).

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.

My view on the matter:

I see game modes as collections of special settings and game features, defined by the game designer. If the code is made properly there will be many modular features and variables for the GD to tweak and combine. What you would consider to be a level is in itself also a combination of variable values, where each item itself is also a set of values.

For reasons of simplicity, balance, marketability and test-optimization the GD limits the availability of all these variables to something very limited.

Games tend to be built around some specific game-mode but if it's well built it can be used for more modes than the initially intended one. I think this is always something to strive for.