I'd say that ground speed should be defined as the speed of the aircraft over level ground, as defined by the rate of change of the position of the aircraft as projected vertically onto the same level ground.
Then again, "vertically" is not constant and usually not "directly towards the center of the earth", nor is "level" usually "perpendicular to a vector through the center of the earth". The geoid is rather uneven, meaning the speed of movement of the vertical projection on the surface is not simply a function of the radii of the geoid surface, the radii of the path of the aircraft and velocity. Constant velocity, constant radii of travel => varying ground speed, as defined in the first paragraph.
I thought I'd throw that in, as the discussion so far was oversimplifying the subject somewhat.
Please stop complicating probably one of the most simplistic definitions in the aviation world!!