Regarding terminology, programming, art, programming, design, etc, are all aspects of 'Game Development', not 'Game Design'.
Game design is only, that - design. The person making the music, or the art or the code, is a game developer, not a game designer.
I'd definitely like to assume you know what you are talking about, but I once argued this exact thing and lost.
Game design is the game development process of designing the content and rules of a game in the pre-production stage and design of gameplay, environment, storyline, and characters during production stage. The designer of a game is very much like the director of a film; the designer is the visionary of the game and controls the artistic and technical elements of the game in fulfilment of their vision. Game design requires artistic and technical competence as well as writing skills.
I don't think neither of us are completely right here .
That quote doesn't say anything about programming, art, music, etc. "Designing the content" in the pre-production phase is purely a descriptive and scheduling task. Likewise for designing gameplay, environments, characters etc during the actual production phase.
From these initial designs, programmers will actually implement the gameplay. Concept artists will create illustrations of environments and characters (which may be iterated on with feedback from the game designer), and 2d/3d artists will then actually create the 2d/3d art based from these concept illustrations.
These artists and programmers will then work alongside the designer to refine and iterate the content according to updated designs.
Within the industry, a game designer is typically a pretty specific title. It's not used to refer to programmers, artists, musicians, project managers, etc... It's used to refer to the people who's full time job is working on the designs. There's then several specializations, such as the lead (organizational), the level designer, combat designer, puzzle designer, etc...
However, anyone at all who works on a game in any capacity, is called a game developer. If you're at a "Game design school", they'll typically teach you to dabble in all the professions and call yourself a game designer, but that's their own choice. It's not how most companies work.
The Disciplines section of that wikipedia page is bunk. UI design and Audio design are not a sub-categories of game-designer, they're distinct professions.