Animation, 3D modeling, level design, concept art, story writing, programming and many different aspect are under the 'Video Game Design' umbrella.

No, those things fall the "video game development" umbrella. Design is a very specific thing.

Ah okay, well can you go into what design is so I can change my mindset/definition, because I had thought it was... Thank you for the correction.

Game design comprises of written work, illustrations and documents regarding how a game is played, it's target market, and everything about it, and is generally tied in with the programming side. Don't re-evaluate your definition of design. Change your definition of game design.