You can't really talk about spatial dimensionality of a game in any other sense than representationally, can you?
The internal representation of a world and the way it's visualized are two different things. You could make a three dimensional world and represent it in the form of purely text based descriptions.
This is in the same way that Dungeon Master lets you move up and down floors. The only reason you can't do that in real-time so to speak, was due to memory constrictions at the time. So the argument that it's only a 2D game is rather off, in my opinion.
Dungeon Master is a two dimensional game, because the dungeons are flat maps that are connected via pits and stairs. Hired Guns has a three dimensional world.
i don't think you can move up and down in Doom
Sure you can, have you even played it?
but no-one would seriosly say its not 3D.
I would, because it's not three dimensional in any way at all. The maps are flat in the sense that you can't have things like rooms above each other, that's why you can actually see everything on the auto map, which would be impossible if the maps were three dimensional.
Then there's the graphics engine which is completely fake 3D. So no, Doom isn't three dimensional.
3D is not about how you can move.
It is to me. It's the world that makes it three dimensional, after all, if I close my eyes the world doesn't stop being three dimensional. What it looks like is completely irrelevant.
Its about whether depth is show on screen, and how this depth varies to mimic the real world as the viewing perspective changes.
Even that isn't actually three dimensional, because it's just a flat surface. And besides, humans can't see in three dimensions or we would be able to see an object from all sides at the same time. We see two completely flat images, and two flat images don't make a three dimensional image. The brain creates a three dimensional approximation of what you see (whether you have one or two eyes), but the brain's visual input is as two dimensional as it gets.
3D is all about the internal representation, and three dimensional visualization would be where you can actually stick your head inside the viewing system so that you can look behind things, etc.
Basically it all comes down to the difference between the way humans normally talk, and how things really are. You say it's three dimensional, even when it's not, because it looks like it's three dimensional.