Allan:

Do you know anything about resultant vectors? If you do it'll help you wrap your brain around the issue.

Let's take a scenario: You are shrunken to model size and you are riding in your plane. Your airspeed is 40 mph. You turn in a constant bank at a constant speed. What's your airspeed? It's 40 mph of course. Now assume it's a breezy day and you execute the same manuveur. What's your airspeed? It's 40 mph. There's no change from the airplane's perspective.

Now stand on the ground and do the same thing. What is your airspeed? It's still 40 mph, but the model will be drifting away from you at the speed and direction of the wind.

The reason for the downwind turn phenomina is perception. You as a ground based pilot can not tell what the aircraft's frame of reference is. You fly it from your perspective. You make corrections to keep the plane in your "box" and not drift downwind. Hence the plane appears to be performing very differently. In reality, though, it is all in response to your inputs.

The inertial picture isn't different because of the wind. It is different because of the pilot. I can express for you the Newtonian equations from either reference if you wish, and show that they resolve the same. I'm lazy and would rather not. Hopefully, this explanation will suffice.