There's a logical programmer who writes the PCs walking around and such... Then there's a scripting programmer who writes stuff like "If you click this button, it will open this door." Then's there a graphic programmer who...? and a level programmer who...? Any other programmer types?

I think you're confusing programmers with asset-makers. There are no level programmers, only level editors. Usually there are very few programmers compared to artists working on a game. You usually have a general-purpose programmer and a graphics programmer working on a title, with various programmers who contribute to parts of the code. If the game has advanced audio, you might have someone specialized working on the audio code.

For the most part, artists create nearly all of what you actually see, from models to levels to the menu. There may be a few exceptions, such as particles.