As far as what the point of categorisation is, well, it provides tools in order to assess issues with a game.
Here's some wood to build the frame of a house. And a pipe wrench. Go!
I have noted, several times, that you need to be more specific about the goal before you figure out what definitions you need. You cannot just define terms, and expect that they'll be useful in general. That's like expecting a tool picked from your toolbox at random to be applicable to a particular home repair job you're trying to do. What issues are you trying to assess? In what games? Or, do you figure somehow your non-purpose-built definitions will be good for all issues in all games? Define yourself a hammer, and every game will look like a nail!
In the world of physics, I might point you to coordinate systems - your choice of system can dramatically simplify, or dramatically complicate, a job you are trying to do. Any flat three dimensional space could be described with spherical coordinates, or rectangular coordinates. Writing down the motion of our local planets in, say, Earth-centered rectangular coordinates is insanely difficult. Doing it in Sol-centered spherical coordinates is far easier. Meanwhile, you'd likely not use spherical coordinates to describe the addition you'd like to build onto your home to the contractor. But it is still just a flat 3D space we are talking about.
Case in point: Forge Theory and the 1999 WotC survey give us two differing perspectives on games and gamers. Different language, defining different things, but both looking at the same phenomena.
Pick the job first, then pick the tools, not the other way around.