[RedDevil]

I would think vortices can (and usually will) be created by disturbances in a local part of the body, so you can have several distinct vortices from both downforce and lift generating surfaces, as well as from disturbances that do not generate net lift or downforce at all.

In the F1 car it is clear that the vortices are a local affair, rotating around a core that originates from the sides of the rear wing.

Two other important vortices (not visible in the photograph) are generated by the gap between the nose and the louvres on the front wing, then tunnel between the front wheels and the nose and hit the barge boards behind the front wheels, forcing them out and helping establish a low pressure area below the car's floor.
The vortex also defeats or at least limits the turbulence created by the front wheels that would otherwise mess up any aerodynamic trickery in that area.

Vortices do not necessarily have anything to do with lift or downforce, but can be pretty useful for altering the flow in a controlled way.