We could go into circulation theory as well if you like but we won't complicate things. Simply put Lowdown is spot on. It is Newton's third law in action. For every action there is an equal and opposite reaction.
Bernoulli's principal is simply a way of quantifying the amount of air displaced by the wing according to Newtons Third law. I.e. for a 2000kg aircraft to fly straight and level 2000kg's of air must be displaced at any one time for the aircraft to remain flying level. Therefore they are both correct.
However according to CASA all you need to know is Bernoulli’s theorem so don't go getting confused by all the rest of the (correct) theories. I believe CASA exams still state that two particles of air that split at the front of the wing meet again at the trailing edge? (May not be anymore, but it was when I did my theory). What a load of hogwash! But again don't worry about it, as the theory works regardless of the fact it is wrong.
As stated in other posts a Split flap increases the amount of air displaced and thus increases lift (does stuff all to the airflow over the top of the wing). It does also increase drag so you need to overcome this to maintain speed (more power) or you can reduce speed and get the 2000kgs of lift and thus stay straight and level. Think about it this way, all an increase in AoA is doing is presenting more of the bottom of a wing to the airflow, therefore it displaces more air down thus creating more lift. You can also increase lift by increasing speed iaw L=0.5(rho)V^2 S Cl . Increasing speed again works with Newton as during whatever period of time, the amount of are displaced increases with speed. Again Bernoulli's equation simply quantifies this.
There are other factors that we can go into but you don't need to worry about them unless you are enjoying the fun of studying for an Aeronautical Engineering degree.
So there you have it, a rather verbose explanation, but just remember that whatever theory you adopt they are in fact all correct (except for the two particle theory - an abuse of Bernoulli's work and Bernoulli never ever said that). However they do all simply come back to Newtons Third Law
QED
Any questions please fire away. I love this Sh*t!!! (yes I am a sadistic Aero Eng...)
Cheers
CB
Edited to add, The top shape of the wing is largely irrelevant. Essentially all the top of an aerofoil is good for is reducing the drag. A flat plate will produce lift (but a massive amount of drag), thus to ensure that our C172 flies with 150HP we have an aerofil shape that controls the airflow across the top of the wing reducing drag so we don't have to fit a merlin to a 172 (although that would make it sound better!). That should put the cat amongst the purist's