Why is everyone talking about changes in the stall AoA when flaps are up or when flaps are down when I've always been taught that the wing will stall at the same AoA?
There's your problem.
You need to be a bit more precise - a wing will always stall at the same AoA
as long as it's in the same configuration. Changing the configuration
will change the properties of the airfoil and alter the lift and drag coefficients and - curves.
Rules of thumb; trailing edge devices (flaps) lower the Critical angle of attack, leading edge devices (slats) increase the Critical AoA, both increase CLmax and both decrease the L/D ratio.