Both explanations are equivalent. The wing forces air down, so by Newton's third law (and momentum conservation), air pushes the wing up. This upward force manifests itself as pressure on the wing since air is well-modeled as a fluid and the air being forced down combines with the air below the wing to form a higher density, high-pressure region. The space the forced-down air left is now less dense and has less pressure, creating a pressure difference between the top and bottom of the wing. Because the pressure on top of the wing is less than the pressure on the bottom, the overall force on the wing is upward.
Another way to say this is that the Newtonian view is the microscopic view--imagining individual air molecules colliding with the bottom of the wing to hold it up. The pressure-based (or Bernoulli) view takes a macroscopic view, in which fluid air interacts with an obstruction and flows around with varying speeds and pressures. The Bernoulli view can be derived from the Newtonian view.
After watching the Sixty Symbols video, I'm understanding the confusion. The image that comes to mind when considering the simple Newtonian picture is incomplete in one sense because it looks like only the air under the wing makes lift. The Bernoulli aspects of the explanation gives the result that air flowing over the top of the wing ends up with a larger downward momentum, and so contributes more lift than the air flowing under the wing. The complete picture not only considers the air that interacts with the wing, but also how that wing-pushed air interacts with the surrounding air.
The simple statement is still true: airplanes stay up by moving air down. The detailed explanation gets into hairy fluid-dynamics math that tells you that wings both push and pull air down (among many other actions).