2.5D is more of a retronym after the age of "full" 3D engines came to be. Before then, we'd call anything 3D that looked 3D, and the lines do certainly get blurred.
Wolfenstein 3D for example is practically a 2D game, you could redisplay it as a top-down shooter reminiscent of early 1980s games (much like the early Wolfenstein games, actually), and not lose anything. There is no Z axis in consideration for any game logic.
Doom, on the other hand, actually does have a Z axis, determining which floors can and cannot be reached, how they are displayed, whether projectiles hit enemies or not, and so forth. The (original) engine and map format don't allow any rooms-over-rooms though, so it is still considered 2.5D -- somewhere in between a 2D game and Quake.
I remember many of the isometric drawn/pre-rendered games with a fixed PoV (i.e. Zaxxon etc) being described as 2.5D back in the day. We knew they weren't exactly 3D but then again, they weren't your typical 2D platformer/side-scroller either. Battlezone was considered a 3D game back in its day so the rendering technology wasn't really the dividing line. It was more a question of if the game environment being navigated was 3D and if it had a reasonably realistic spatial representation on the screen.
Zaxxon was definitely 3D. You scrolled forward, could move side to side, and could also change height (and you had to, to avoid obstacles and shoot enemies)
To make things more confusing, 2.5D these days is also often used to describe (actually) 3D-rendered sidescrollers, like the New Super Mario Bros. games.
A "first person engine" could be ray-casting on a 2d map like the original Wolfenstein, or on a 2.5d map like the original Doom, or actual 3d like Quake and everything since. It's a broad term.
The fact it isn't called a 3d engine, to me as an old person who played Doom on its release, makes "first person engine" a more appropriate. name
I thought 2.5D was used for 2-d rendered games that had depth, like zelda: a link to the past (very much a 2d game where you can jump down cliffs, but not climb up them).
2.5D started with Doom and it refers to games where you can move on Z axis, but can never be at the same XY point at more than one Z. That is you can't walk under or over some other path.
I thought 2.5D was the term used for 2D assets used in a way to appear 3D.
Pardon my lack of familiarity with game engines if I miss-understood this.