It basically means everything is happening in the shaders, not in geometry. There has to be some vertexes though, and the minimum you can have to cover the screen is two triangles.
That is incorrect. You can cover the screen with a single triangle ... just make it big enough. The corners get clipped and the middle region of the big triangle will cover the screen.
True, both approaches are equally valid and being used. Every once in a while some gfx haxors also spend baffling amounts of effort on humorously "benchmarking" both approaches against each other.. ;)