For example, from Newton one can state that the whole lifting force F_lp acting on the plane is opposite to gravity force F_gp acting on the plane, so plane flies horizontally (in the simple case): F_lp + F_gp = 0. Next, from third law of mechanics the lifting force F_lp acting on the plane is opposite to F_da force acting on air downward: F_lp = -F_da. From the second law of mechanics force F_da acting on air causes the air momentum to change: F_da = dp/dt, which means that either the plane pushes a lot of air down or the plain pushes air down really fast, just like article says.

Looking into the picture from Bernoulli perspective, we can talk about flows around the wing. If say we have a wing which is flat underneath and curvy on top, the air is initially pushed upwards until the top point of the wing is reached. From there air continues along the wing profile, which goes down, so air has a chance to accelerate - this is effectively an empty space which becomes available to the air, and air goes from points of higher pressure to points of lower pressure. The air speed increases, the pressure remains less than in (unchanged) air below, so we're getting the Bernoulli effect - lower pressure above the wing.

If the plane is upside down, it won't fly horizontally. It has to fly somewhat upwards - in which case flows on both sides of wings change, with net effect the same - the air is pushed down, pressure under the wing is higher than above, both Newton and Bernoulli explanations work.