Sit in the aeroplane on the ground, parked with the brakes on, but facing into a 30knot wind. Ground speed zero, air speed 30knots and the ASI will actually indicate that. So what you need to understand to make your confusion go away is that when the wind is blowing, an aircraft that is fixed to the ground actually IS MOVING (relative to the air).
Fly at 50knots airspeed into the same 30knot headwind and the ground speed will be 20knots. The aeroplane doesn't care what the ground is doing, only the air over its wings.
The only time you need to worry about ground speed is when navigating - it doesn't affect the aerodynamics.
How IAS and TAS relate to each other is a different thing altogether.
Hope that helps - and by the way your conveyor belt analogy is perfectly good.