The NASA quote...
The third law states that for every action (force) in nature there is an equal and opposite reaction. In other words, if object A exerts a force on object B, then object B also exerts an equal force on object A. Notice that the forces are exerted on different objects. The third law can be used to explain the generation of lift by a wing
...can be used to explain lift by a wing. The wing accelerates air downwards and forwards (by applying a downwards and forwards force to it), causing the wing to experience an upward (lift) and backward (lift drag) force. The more you accelerate it (by increasing the angle of attack), the greater the forces are.
What is actually going on when an aeroplane flies is enormously complicated. In order to understand it, we build simplified models. There are no 'right' or 'wrong' models; however some models explain certain aspects of flight better than others. Therefore, it is sometimes easier to think of an aeroplane as having thrust, drag, lift and weight, and sometimes it better to think of it as other forces (for example splitting the lift force into a horizontal and vertical component when the aeroplane is turning).
I think you should be comfortable with both the 3 and the 4 force model (and perhaps models with a few more forces as well to handle turning and spins and so on), but don't think that one is correct and the other is somehow wrong.
A good model should:-
Be simple as possible
Be able to explain observer results
Be able to predict future results