The distinction you are making is not inherently necessary. It depends on the game in question. Here are some basic terms and how they would apply:
solitaire — any of various card games that can be played by one person
"Solitaire" has been adapted for usage in board games that allow one player to play on their own but is still technically considered exclusive to card games. It is really only used for "puzzle" or "challenge" style games where you are trying to accomplish a particular goal based on a set of rules (i.e., the game). It would not be remotely appropriate for running through an FPS on your own just to learn the details of the map.
single-player — A single-player video game is a video game where input from only one player is expected throughout the course of the gaming session. "Single-player game" usually refers to a game that can only be played by one person, while "single-player mode" usually refers to a particular game mode that is designed to be played by a single player, though the game also contains modes that can be played by several players simultaneously.
There is no suitable distinction between a single-player game that refers to competing against AI versus a single-player game similar to solitaire. Multiplayer is merely the opposite of single-player:
multiplayer — A multiplayer game is a game which is played by several players.
Which brings us to the definitions of various players:
player — A player of a game is a participant therein. The term 'player' is used with this same meaning both in game theory and in ordinary recreational games. Normally, there are at least two players in a game, but one-player games exist and are collectively known as solitary games (such as the Solitaire card game and many video games).
Most players are human players but it is technically correct to refer to any active, independent competitor as an "AI" or "computer" player. The key distinction between an AI "player" and just some "AI" built into the game for the purposes of challenge is whether the participant fulfills the same role as a human player would.
What this means for the terms of single-player and multiplayer is that the game can be designed to support built single-player and multiplayer modes (see above) and those player seats can be filled by human or computer players.
Typically, however, the menu options in games need to distinguish between a game full of human opponents and a game against AI opponents. This terminology is very game specific and although the industry has some conventions there is no technically correct way to label things. The conventions I see most often:
- A computer player added to a game with multiple human players is refered to as an "AI" or a "computer player" (amongst other terms; e.g. "bot"); the mode is still called multiplayer.
- A single human playing against computer players is typically referred to as a single-player mode or practice mode unless that player simply chose to fill up all of the multiplayer seats with computers. Then they are still playing in multiplayer mode. but this is sometimes referred to as a single-player game.
- A single player playing against a set challenge is now commonly referred to as single-player mode or "campaign mode" (amongst other terms; e.g. "story mode").
To directly answer your question:
a human player with no opponents
This is referred to as single-player or solitaire, depending on the content of the game.
a human player with a computer/AI opponent
This is referred to as single-player or practice or multiplayer depending on how the game was started. If it is a unique game mode that is completely seperate from the normal multiplayer mode and you have a standard singleplayer mode then you will need a term similar to "practice mode".
If "practice" is unsuitable you could use any of the following:
- AI mode
- Bot mode
- Vs Computer (and rename multiplayer to Vs Players, Vs Humans, etc.)
multiple human players
The best term for this is multiplayer. Other terms for it exist but are only really used when more than one multiplayer mode needs to be distinguished.