A unifying concept that defines the atmosphere, visuals, and story
The theme is the unifying concept that determines the atmosphere, context, or subject of a game. This can be abstract, such as fear, or more concrete, like a pirate adventure. The theme sets the direction of the art style, music, and the story.
In puzzle design we use theme to encompass all the aspects of a puzzle that are neither a device nor a concrete puzzle piece quality nor mechanic. In this sense, the theme is effectively everything that could be stripped away without altering the core puzzle design. That same puzzle can be represented in a different theme without altering how the player would go about solving it.
This isn't to say the theme is merely aesthetics, or that they play a secondary role in puzzle design. The theme is used to convey information to a player in a way they both understand and appreciate. The theme ties abstract devices to real world situations.
Assume everything in this reference is a working draft, there's prone to be some mistakes and inconsistencies. I figure it's best to publish and get feedback rather than write for years in secret. The terms will change, the structure will shift, and the bugs will be chased out. It'll take a while.