Does ground speed represent the horizontal speed measured at sea level?
(Source)
Ground speed
Ground speed equipment on-board aircraft measure the speed against the ground. Except in the case shown above where distance information is derived from a ground station (DME station).
Curvature
Ground speed is not derived from true airspeed. When using GPS, or an inertial navigation system to measure the ground speed, it will be the speed across the ground. It is the rate of change of the latitude and longitude, so you don't need to include the curvature.
Once an aircraft (or crew) know the ground speed and TAS, the wind information is then known (how fast the mass of air the aircraft is in is moving).
True airspeed
True airspeed is the speed inside a mass of air. In a zoom climb in an F-15, TAS will be pointing almost the same direction as the [upward] nose of the F-15.
If the mass of air is moving due to head- or tail-wind, then this wind speed is subtracted from (or added to) the horizontal TAS to get the ground speed. (But again, this is not how ground speed is calculated, because there is no direct measurement for the horizontal TAS.)
Vertical speed is how fast the plane is climbing/descending. It's measured by sensing the rate of change of the static air pressure.
RE: edit

I'm simulating aircraft movement along a straight trajectory by constant speed.

This is not how airplanes fly in real life. But for the sake of your model, and if you don't have wind simulation, then the known green distance and the time it takes to move from A to B will do just fine to compute the ground speed.