So presumably we've got tropical cyclones (hurricanes, typhoons, etc), sub
tropical cyclones, extratropical cyclones (which presumably may or not be
the two previous) and mid-latitude cyclones (depressions which may or may
not be formed from the three previous or have absorbed the three
previous) -- argh. Better'd do some reading.

....and just to throw in a bit more you've got the Polar Low,sometimes called an 'Arctic Hurricane'
(can sport an eye feature),probably classifiable as a 'mesoscale extratropical cyclone',but have
been called 'polar mesocyclones'.
then there's 'sub tropical mesoscale extratropical cyclone' eg 'Mediterranean Hurricanes',

--
regards,
david