This will be one of the more challenging super-categories to organize. For now I'm going to quickly list all the types of organizations I can find, and with others hopefully a scheme will emerge.
I'm proposing that the categories for each organization cover the following basic topics: its "birth", "death", "location", "legal status/organization", and "purpose". Note that legal status & purpose can be combined in various ways -- for instance, "professional associations" are both within legal structure (non-profit) and activity.
Within the above proposal, most of the "organizations by field" are "subject" categories and that tree is being filled in -- Category:Organizations by subject. I believe that Category:Organizations by kind is better thought of as Category:Organizations by activity and Category:Organizations by legal status, and hopefully those terms are a bit more descriptive than "kind". Category:Organizations by geography is clearly the location category.
Thoughts?