Unfortunately, those figures don't really explain anything. In particular, Figure 4 doesn't explain why the red arrow next to the propeller is drawn pointing to the left (thrust) instead of to the right (drag). I also don't understand why the relative wind is still stated as zero knots when the cart is alleged to be traveling faster than the wind.
I'll try the words part again and save it in notepad this time.
In box 1: the cart is stationary. The 10kn wind begins pushing the cart foreward, because there is drag which acts in the direction of the relative wind. This part happens whether there is a propeller strapped to the back or not.
In box 2: the cart begins to move (because the wind pushes the cart). When the wheels start turning, they mechanically drive the pusher prop to create thrust. At this point, the cart is still traveling slower than the wind, so the drag force is still an accelerating force, as well as the thrust created by the prop which is only turning because it is mechanically connected to the wheels.
In box 3: the cart is traveling exactly as fast as the wind. Because the relative wind is 0, there is no drag, but the propeller is still producing thrust accelerating the cart. The cart does not get stuck at this point or begin to slow down because if you sum the forces, the only force acting on the cart is the thrust from the propeller. The propeller does not stop turning because it is mechanically driven by the wheels, NOT turned by the relative wind. There is no drag therefore no force equal and opposite to the thrust therefore sum of forces says the cart must continue to accelerate.
In box 4: the cart accelerates faster than the wind. There is now relative wind acting against the motion of the cart, so now there is drag acting as a decelerating force. I can't tell you exactly what speed it will happen, but it must happen at a cart speed that is greater than the wind speed in order to create any decelerating drag force, let alone a drag force that is equal and opposite the thrust created by the propeller.
Richard, you said "Figure 4 doesn't explain why the red arrow next to the propeller is drawn pointing to the left (thrust) instead of to the right (drag)." - Its the same as an airplane. When you're flying, there is a relative headwind equal to your airspeed acting against the direction of travel (airspeed indicator), which does create drag on the airplane, but because the prop is being turned by the motor, it continues to produce thrust (the thrust vector points in the direction of travel). In the cart, the propeller is not being turned by the wind, its being mechanically driven by the wheels, and no matter which way the wind is blowing, if the wheels are turning, the propeller is producing thrust.
You also said: "I also don't understand why the relative wind is still stated as zero knots when the cart is alleged to be traveling faster than the wind." - you didn't examine the figure closely enough. In box 4, the relative wind is stated to be greater than zero left to right, or against the direction of travel. I didn't put a number there because I can't tell you exactly what speed the cart will have to be traveling, only that it must be greater than the speed of the wind in order to produce drag equal and opposite to the thrust.
Here is a thought experiment. Suppose there is no propeller, and you have a very light cart with wheels that produce little friction (negligible). There is a tail wind. The tailwind creates drag and pushes the cart forward. As long as there is a relative wind, there will be drag pushing the cart forward. Drag is the only force acting on the cart. The cart will continue to accelerate until sum of forces equals 0. Since drag is the only force, the cart will continue to accelerate until there is no drag, or until the relative wind is 0. That happens when the speed of the cart equals the speed of the wind. Therefore, a cart with no propeller will be pushed exactly as fast as the wind.
With that in mind, take the cart with no prop traveling exactly as fast as the wind, and add on the propeller mechanically driven by the spinning wheels of the cart. The cart was at steady state, wheels spinning, traveling exactly as fast as the wind, now turn on thrust from the prop and what happens? The cart must speed up because sum of forces no longer equals 0.