I think a 5 page requirement, min 3 perhaps, will help as proof of concept/dedication/interest. Or if you wish to add a sign-up section for fledgling games to distiguish them from those that arnt.
Suggested edits
Suggestion Games
Games wherein all players suggest the action
taken for a single entity. for the turn.
Single is not the defining quality, just a commonality. All players choosing is. As you said, gameplay not theme ect.Competitive Games
Games wherein players
suggest actions for different competing singular entities. take turns to win a goal that is at odds with one or more players.
Competitive games can be abstract. Connect 4 for example. Move away from the entity concept, it's a theme not gameplay.Cooperative Games
Games wherein players
suggest actions for several cooperating entities. take turns to win a goal that is the same as or compatible with one or more players.Faction Games
Games wherein players are split into playing as different competing factions.
Is this really a large group(is this basically just the middle ground of the previous two?)? Generally any cooperation qualifies it as that type.Roll to Dodge
Roll to Dodge games.
Not needed.Other Games
Games that do not fit into the above categories.
A few more things. While goal is a component, I would not even say it's the main component of gameplay for games. There's some things to take note of. So far every forum game I've ever seen is a take turn game so that's a default.
I think sorting by goal is actually good and probably the only way to do it even though it's not actually gameplay, as the next best thing is gameplay elements and games can have multiple of those. I think sorting by goal rather than theme is a good call, so stick to that. For reference here are some actual gameplay tags.
puzzle
risk-reward
tactical/strategy
random