"In-Cavern", or "IC", is a term that was coined during the lead-up to the original release of Uru in 2003. It was co-opted from the term "in-character", which is used in traditional table-top and live-action role-playing as a way to express treating the circumstances your character finds itself in as though they were real, and as though you were your character.
In traditional role-playing scenarios, in-character behavior requires donning a persona and acting in accordance with that persona's character, which may be drastically different from your own behavior. For example, in Dungeons and Dragons, one could choose to play as an evil character even though you are not inherently evil yourself, and be expected to "play" your character as they would behave, not as you would.
In the Myst franchise, the term In Cavern has taken on a slightly more reserved meaning. In Cavern does not require players to don a persona or play a character. It merely indicates a willingness on the player's part to treat the game's world as though it were real. From an In-Cavern perspective, the D'ni Cavern, the DRC, and all of the trappings of Uru are real places. Perhaps confusingly, The Myst franchise itself exists within this In-Cavern world, as the DRC provided Cyan Worlds with the reference materials needed to develop the Myst games. From an In-Cavern perspective, the events depicted in those games actually took place, and the games simply enable players to don the mantel of an unspecified historical figure within the game world's history to re-enact past events.
Cyan Worlds' involvement in Uru is more difficult to quantify from an In-Cavern perspective, as the DRC is responsible for releasing the various Ages and areas available in the game, and these areas are "real", not part of the Myst franchise proper. Some have argued that The Path of the Shell was a Cyan creation based on further materials provided by the DRC following the closure of the Cavern, while others counter that The Path of the Shell should simply be another In Cavern experience that should not be treated as a game within the Myst franchise. Cyan's official stance on this point is unclear, though the various alterations made to the content in Path of the Shell when it was released in Myst Online: Uru Live is used by some as evidence of the former possibility being true.
Some players have taken this term and applied it to their gameplay style exactly as used in role-playing games (in-character), creating a character and playing someone other than themselves, with mixed results. Because of the "you are you" pitch used for Uru's gameplay style, players tend to directly associate the avatar on-screen with the player behind it. Typically, positive characters (those you could classify as "good" or "neutral" in Dungeons and Dragons, for instance) have been either well-received or simply ignored by those not interested in role-playing in Uru. If the character being played is abrasive to other players, however, the person running that character has historically taken a considerable amount of flak for their character's behavior. Whether this is warranted or not is a subject for a class in game design and social interaction, and extends well beyond the scope of this article.