I could see an argument being made for 'fake 3D' pretty strongly here. DF Retro has a few videos comparing various consoles, and one of the things that the PS1 and Saturn both got wrong, IIRC, was that textures would warp*, because the 3D look was being provided by affine transforms (as I understand, I may be misremembering the details), rather than perspective correct rasterization.
So, like, the vertexes are in the right spots, but the texture mapping is off. So it kinda depends on if you need texture mapping to be perspective correct to consider a given example of 3D "real" vs "fake". Dark Forces also has perspective warping that happens in it's engine when you look up/down, as an example.
Exactly, straight lines on a texture should obey perspective (think rows of bricks on a wall). I think this wasn't as widely understood back then, but notably, id software already got it right a few years earlier in the original Wolfenstein.
Great link. The artifacts are especially apparent when the walls are near the camera, for example at this timestamp: https://www.youtube.com/watch?v=VutzIK3DqZE&t=398s And note that here, the seemingly flat walls are already heavily subdivided into smaller polygons to lessen the impact of the problem. (Why they wouldn't then make use of those additional polygons to include more geometry details for free is beyond me, however.)
The way you've worded this makes it sound as though programmers of the era were somehow unaware of perspective mapping, which I can assure you is not the case at all. Affine texturing was a tradeoff of performance, nothing more.
Wolfenstein's correct texture mapping is due to it using raycasting in a 2D plane and rendering scaled vertical strips of texture, which just happens to be perspective correct because you only ever have surfaces at 90deg angles in the vertical.
As to why those extra triangles weren't used for detailing, it's likely because that would take up extra level data. Tessellation of an existing triangle into smaller triangles doesn't.
Wolfenstein and Doom are 100% correct because Carmack 'cheated' by deciding to never look down/up or draw slopes :-) so whole game is drawn with 'lines of constant Z'. Or as they put it
Chris Hecker (Microsoft/Maxis/etc): that's a classic Carmack thing which is like Fuck those general problems, Im gonna solve this other problem perfectly
John Miles (ORIGIN/Miles Design/etc): Its all about not doing the math, we were still at a point in time when you won by not doing the math
So, like, the vertexes are in the right spots, but the texture mapping is off. So it kinda depends on if you need texture mapping to be perspective correct to consider a given example of 3D "real" vs "fake". Dark Forces also has perspective warping that happens in it's engine when you look up/down, as an example.
* https://youtu.be/VutzIK3DqZE?t=341 DF Retro analyzes ports of Tomb Raider