I’ve said many times that I don’t find the threefold model of role-playing games very useful. I do think things in that vein are needed, though. The most important question when selecting an RPG system isn’t something like genre; it’s play-style. Here’s something I think might be useful.
On one axis, we have “Let’s create a story!” on one end and “Let’s see what happens!” on the other end.
On the other axis, you have mechanical complexity. This could also (depending upon context) be a measure of how much the game or the participants expect to “stick to the rules”. The “stick to the rules” attitude is often associated with more mechanical complexity, but I think the correlation can be loose.
I thought it would be cool to have pithy abbreviations like AD&D alignments. (NC for “narrative-driven + complex mechanics”. CS for “character-driven + simple mechanics”.) But...shrug.
It is really important that both axes are continuums, although I’m sure everyone would ignore that it practice.