What is the time derivative of the angle of attack?
Assuming that there isn't an error in the books you've read:

Both formulas are true. It's true that

$$\frac{d\alpha}{dt} = \frac{w'u - wu'}{u^2 + w^2},$$

and it's also true that

$$\frac{d\alpha}{dt}= q + \frac{Z}{mu}.$$

From this, we can conclude that

$$\frac{w'u - wu'}{u^2 + w^2} = q + \frac{Z}{mu}.$$

Both of these two formulas describe the way that the angle of attack changes, but they describe it in different ways.

I'm not sure exactly how the $q + Z/mu$ formula was derived, but it seems to be related to the fact that angle of attack is related to pitch and the flight path. Specifically, for flight with zero bank angle, the angle of attack is the pitch angle minus the climb angle. It looks like the $q$ term comes from the rate of change of pitch, and the $Z/mu$ term comes from the rate of change of climb angle.