A number of circuits have been explored and classified with terms like cold-clippers, diode clippers, warm distortion, etc. These terms relate to circuits that are known to produce certain OD tone qualities. Discussions of common circuits are found in places like Rob Robinette's
overdrive web page, Internet forum articles, etc. Well known examples include an unbypassed high-bias resistor (produces low bias current) gain stage (a cold-clipper) and a direct-coupled cathode follower that produces "warm" distortion. It's not hard to find examples of such categories, however, they are all only points in a huge space of possibilities where relatively small changes to component values or circuits can sound very different.