I think the equations are not referred to the same axys.
The first equation refers to the angle of attack, the angle between both components of the velocity refered to the body axys (x along the body), using a non-inertial frame (as moves with the missile) . Well, actually following some comments, is not actually the angle of attack, it will be the angle of attack in the specific case of v=0.
The second equations looks to be referring to the rotation of the missile referred to an inertial frame of an external observer (for example, Earth), so actually, it looks like the actual movement of the body frame refered to that observer in a specific rotation component .
Imagine an example, that I am able to provide the right thurst to the missile in such a way that I keep u, w and q constant rotating in a circle.
In this case, the angle of attack will be constant and its increase 0 with time but the second alpha will be increasing/decrasing with a constant increase/decrease, also this change will be linked to the circle rotation.
I think you can also make the math and relate both equations (you will need more components!) taking care of inertial forces.
Final comment, as you are introducing as well a second term in teh second alpha equation, it really looks that it is an specific case in the movement of missile.