From an aerodynamics viewpoint, wing are is merely a reference area used to determine the value of the nondimensional coefficients, for instance, Drag force = Cd x density/2 x reference area x velocity squared. For most airplanes, this includes the fuselage area between the wings. So the "right" wing area is the number the aerodynamics used in their calculations. The fillets, the round tips, flap area, wing position all may affect how much lift, how much drag, etc, but the wing reference area is just a number. It is usually close to the projected gross area.
One example of a gross difference in wing area and reference area is the F-14. The wing reference area is the area defined by the sweep of the wings in the forward position and included that projection to the centerline. Ignores the wing strakes, center body and nacelles completely, even though all, including the forward fuselage, generate lift. (and drag, etc).