What is the physical meaning of Mean Aerodynamic Chord?
The mean aerodynamic chord is only identical to the mean chord for rectangular wings. For tapered wings it is slightly longer.

Why? Because the mean aerodynamic chord is the mean chord of a rectangular wing that has the same pitch characteristics as the "real" wing. Pitch characteristics need to include pitch damping, and pitch damping grows with the square of the local chord. That is why the formula for the mean aerodynamic chord divides the square of the local chord by the wing area:
$$\text{MAC} = \int_{y=-\frac{b}{2}}^{y=\frac{b}{2}}{\frac{c^2}{S}}dy$$
Here $b$ denotes wing span, $y$ the spanwise coordinate and $S$ the wing area. Since the chord is squared, deeper sections of the wing are overrepresented in the result. The resulting rectangular wing will have a larger area than the original, tapered wing but the same pitch damping! In case of a delta wing, MAC will grow to be ⅔ of the root chord, and for an elliptical wing it will be 90.5% of the root chord.

The MAC has been invented to convert arbitrary wing planforms into much easier to calculate rectangular wings. By doing all calculations on the correctly sized rectangular version, the more complicated calculations on the real one could be avoided. However, this works only up to a point:


if you want to calculate lift, you need a wing of equal area.
if you want to calculate induced drag, you need a wing of equal span.
if you wand to calculate pitch motion, you need a wing of the correct MAC. This rectangular wing can either have the same area or the same span as the real wing, but not both together, unless the real wing is rectangular, too.


You are right, the two definitions you quote are not compatible. The rectangular wing cannot have the same span than the real one, or its area and all associated forces and moments would be higher.