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You're right; there aren't strict subsets and we don't know all the different possible piece types. When I heard multi-positional, I was thinking of something like a Wall that takes up two squares. Perhaps you mean a piece that can move from multiple positions as origin, viz Fourriere's Wizard. This brings us to a new division, variable-destination pieces and invariable-destination pieces. An independent-destination piece could always move to the same squares. Dependent-destination pieces move to different squares based on certain variable statistics. The most obvious dependent pieces are those that have movement trace a path from their current square. An alternative piece would be the Loner, which moves to any squares that have a piece density of <25% as calculated from an average of all possible 3x3 rectangles on the board. It becomes trivially more powerful toward the endgame.