As to whether a given engine "pushes" or "pulls" - simply put a stressmeter on the connection between engine and airframe and determine whether the stress is in tension ("pulls") or compression ("pushes.")
You can fly with a barn door for a wing - with enough power. Water skis provide lift with little or no Bernoulli input. Just downwash and AoA.
However, Bernoulli effect makes the process much more efficient, and makes the difference between being able to fly, and being able to fly functionally (with a useful amount of payload capacity and range).
With barn doors for wings, Orville and Wilbur would have needed closer to 1,200 hp to get off the ground, rather than 12.