I think you try to pigeon hole things too neatly.
Consider that, however you might want to explain things, an aeroplane doesn't fall down because, quite magically, it throws air downwards and, as a result, develops a force (lift) upwards.
Different explanations are useful for different purposes but all arise from the above observation.
If you want to get a handle on what really is going on, you need to read up on circulation theory. The wing generates a circulation and that is an observable thing experimentally. The easiest way to see what that does is to look at spinning cylinders in a tunnel or tank. A search on the Magnus Effect will bring up interesting sites for you to review .. you don't really want to play with the mathematics, I suggest, unless you have nothing better to do this evening ...
The pilot doesn't need to know anything much about this level of detail .. but, if you are interested, then go for it.