You’ve got a solid definition for “depth.” Can you give similar definitions for “fairness” and “challenge”?
Fairness is perceptual, people have differing standards of fairness. In single player games fairness is typically a matter of accurately communicating to the player what they’re engaged in, preventing them from getting blindsided, or letting them know that they may get blindsided and how to prevent it. Fairness is typically perceived as a matter of clarity. For that purpose many single player games even overcorrect, adding extra buffers or leniencies to tilt things in the player’s favor, because player perception is not always perfect, and they can miss an input in the space of a frame or two and think they got it when they didn’t. Many games make enemy hurtboxes smaller for this purpose.
In multiplayer games, there’s a similar impetus on communicability for fairness, but more than that it comes down to fairness between players and there’s even more widely varied standards there. This is typically a matter of balance, but players can perceive a lot of things as unfair, leading to scrub mentality. They can get hung up on individual moves that they think are unfair. They can get hung up on other players being better than them. They can get hung up on which characters are good or bad. It’s a mess. Is it fair that players can master certain techniques and win way more consistently? Is it fair to use unintended tricks? Is it fair to use unreactable mixups? Is it fair to have randomness affect the outcome of a match? Is it fair for a player who puts in less time to beat someone who puts in more time? Is it fair that some characters are better than others? A lot of these questions will be answered differently by different people.
Fairness means treating players without bias and preventing injustice. In Singleplayer contexts, preventing injustice is a matter of figuring out what players perceive as injust or imposing your definition of just upon them (“average human reaction time is about 15 frames, if this move’s startup is 15 frames long, you have no right to complain”). Ultimately you’re working to satisfy your players, so you need to be in tune with their perceptions to make them feel like the game is fair.
Challenge is a lot simpler. It’s a task that one is inconsistent at completing, where consistency is dependent on the individual’s skill. It may be used to refer to a specific instance of a difficult task in a game (eg. a reaction challenge).