This just seems like opposition for the sake of binary thinking.

Quite possible, but it refers to the simple political compass; which by the same definition is never accurate (or, binary thinking need not be wrong if you accept that it grossly simplifies).

If we add an element of competition to Minecraft, suddenly we have something that breaks your system – and there are PVP mods for Minecraft that form competition in dozens of ways. Likewise there are PVP modes for Factorio

I've seen Factorio, but not PvP. On what is PvP based- real time and you compete who will have more resources and / or destroy the opponent? The toy aspect of Factorio is then lessened as it now has a goal and measurement, whereas before we could conversely say it had only a direction. With goal and measurement it becomes a game- it is a system of rules the players compete to navigate in the most efficient manner.

The question is what is the focus of the game; if you had to grossly simplify it to find the only the single most important thread (even if that thread were a non-thread between threads as in opposed mechanics: you must earn and spend $ becomes you must manage $).

Weird combinations do happen, but what is the whole? In our context, the primary point of such distinctions would be of course to create interesting categorizations that stimulate new ideas.

If you really like this architecture, you would be far better served in presentation and contemplation by making each trait a scaler rooted at zero. A specific instance of a game then has some portion of all four of your indicators and rather than two oppositional axes, you would have four columns in a bar graph or radial distances of a polygon.

You mean this?