UPDATE:

Okay, I just read further down in the comments of the OP, so now I have a better understanding of the context: from what it sounds like, the game mode is almost the same in all cases, it's mainly a question of the number and type of opponents that you are facing (0-to-# and computer vs. human).

Given that, it would probably make more sense to focus on that aspect of the game . . . maybe:

1. Solo
2. Player vs. Computer or Player vs. AI
3. Player vs. Player

I would avoid going the single-player vs. multi-player route, just because, there is often an implied difference in how the game is played, when using those two titles.

Here is how I find it is being split up these days (for the most part):

1. Campaign or Story mode
2. Multi-player mode
3. Multi-player mode

The thing about the "bots vs. AI" discussion is that, for the most part, they are both the "multi-player" experience . . . the game mode focuses around multiple players playing against each other, either in teams or individually. Whether or not those "players" are controlled by humans or the computer is simply a separate detail of the experience.

Of course, there are games that have broken this dynamic . . . Brink, for example, was a single [primarily multi-player] experience, that used a mix of human players and bots (based on player availability), and also followed a campaign story, as each round completed, but I wouldn't use those exceptions as guidance, unless they apply. :)

UPDATE:

Okay, I just read further down in the comments of the OP, so now I have a better understanding of the context: from what it sounds like, the game mode is almost the same in all cases, it's mainly a question of the number and type of opponents that you are facing (0-to-# and computer vs. human).

Given that, it would probably make more sense to focus on that aspect of the game . . . maybe:

1. Solo
2. Player vs. Computer or Player vs. AI
3. Player vs. Player

I would avoid going the single-player vs. multi-player route, just because, there is often an implied difference in how the game is played, when using those two titles.

Original Answer

Here is how I find it is being split up these days (for the most part):

1. Campaign or Story mode
2. Multi-player mode
3. Multi-player mode

The thing about the "bots vs. AI" discussion is that, for the most part, they are both the "multi-player" experience . . . the game mode focuses around multiple players playing against each other, either in teams or individually. Whether or not those "players" are controlled by humans or the computer is simply a separate detail of the experience.

Of course, there are games that have broken this dynamic . . . Brink, for example, was a single [primarily multi-player] experience, that used a mix of human players and bots (based on player availability), and also followed a campaign story, as each round completed, but I wouldn't use those exceptions as guidance, unless they apply. :)

Here is how I find it is being split up these days (for the most part):

1. Campaign or Story mode
2. Multiplayer mode
3. Multiplayer mode

The thing about the "bots vs. AI" discussion is that, for the most part, they are both the "multiplayer" experience . . . the game mode focuses around multiple players playing against each other, either in teams or individually. Whether or not those "players" are controlled by humans or the computer is simply a separate detail of the experience.

Of course, there are games that have broken this dynamic . . . Brink, for example, was a single [primarily multiplayer] experience, that used a mix of human players and bots (based on player availability), and also followed a campaign story, as each round completed, but I wouldn't use those exceptions as guidance, unless they apply. :)

UPDATE:

Okay, I just read further down in the comments of the OP, so now I have a better understanding of the context: from what it sounds like, the game mode is almost the same in all cases, it's mainly a question of the number and type of opponents that you are facing (0-to-# and computer vs. human).

Given that, it would probably make more sense to focus on that aspect of the game . . . maybe:

1. Solo
2. Player vs. Computer or Player vs. AI
3. Player vs. Player

I would avoid going the single-player vs. multi-player route, just because, there is often an implied difference in how the game is played, when using those two titles.

Here is how I find it is being split up these days (for the most part):

1. Campaign or Story mode
2. Multi-player mode
3. Multi-player mode

The thing about the "bots vs. AI" discussion is that, for the most part, they are both the "multi-player" experience . . . the game mode focuses around multiple players playing against each other, either in teams or individually. Whether or not those "players" are controlled by humans or the computer is simply a separate detail of the experience.

Of course, there are games that have broken this dynamic . . . Brink, for example, was a single [primarily multi-player] experience, that used a mix of human players and bots (based on player availability), and also followed a campaign story, as each round completed, but I wouldn't use those exceptions as guidance, unless they apply. :)