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dtgreene: There's actually a different way to define the dimensionality of a game, one that doesn't relate to graphics at all.
* A 2D game is one where each entity's position is described by 2 coordinates.
* A 3D game is one where each entity's position is described by 3 coordinates.
* One special rule: Only coordinates for the current map count, and not those that indicate which map is used.
That definition would make many games that are visually 3D be considered 2D. Just about every blobber for one...
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dtgreene: Not familiar with Doom, but is it possible to be on top of an enemy, or to avoid collisions by jumping over things? Or to walk over an overpass while enemies move below? If the answer is yes, then it's likely 3D; otherwise it's likely 2D.
The answers are 'no', 'no' and 'no', respectively. Doom is actually a 2D game, but I didn't even include it in my list of FP 2D games.

The simple definition is whether or not the entire game world is internally represented and processed in three dimensions.

Referring back to your point about relating to graphics, three-dimensionality doesn't even have to manifest graphically. Technically, a version of Super Mario Bros. that is fully 3D yet with the mechanics remaining exactly the same, and carefully rendered to appear exactly as the original game, would be a 3D game. From a marketing perspective there would be little point in referring to it as a 3D game, but the fact that it is would certainly be important from a developer's/modder's perspective, as it would determine what is easy/possible in terms of modification.

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dtgreene: There's actually a different way to define the dimensionality of a game, one that doesn't relate to graphics at all.
* A 2D game is one where each entity's position is described by 2 coordinates.
* A 3D game is one where each entity's position is described by 3 coordinates.
...

My definition also works for games with other numbers of dimensions. One could argue that many incremental games are 1D, while it becomes possible to imagine a 4D game with the definition I give here.
That's just it: It's your definition. You bring up an interesting subject for discussion, but that doesn't have anything to do with the concepts of 2D and 3D games as they have been understood for over a quarter century. I was just responding to your implication that there are no first-person 2D games, as a person would normally understand the term '2D'.