I would like a clarification on bank angle and how its different from roll angle with respect to to fixed wing aircraft. It is my understanding that the bank angle is a result of rotating the aircraft body to the stability frame, implying that if the angle of attack $\alpha$ and the side slip angle $\beta$ are zero then, and only then the bank and angle and roll angle are the same. Based on Stevens and Lewis Aircraft Control and Simulation (which doesn't define bank angle) the rotation from body to stability(wind) frame is given by $$ C_{w-b} = \begin{bmatrix} &\cos\alpha \cos\beta & \sin\beta & \sin\alpha cos\beta \\ &-\cos\alpha\sin\beta & \cos\beta & -\sin\alpha\sin\beta\\ &-\sin\alpha & 0 & \cos\alpha \end{bmatrix} $$
Therefore, the bank angle $\mu$ I think is given by: $$\mu \triangleq \begin{bmatrix} &\cos\alpha \cos\beta & \sin\beta & \sin\alpha \cos\beta \end{bmatrix} \begin{bmatrix} \phi \\ \theta \\ \psi\end{bmatrix} $$
where $\phi,\, \theta,\, \psi$ are standard aerospace Euler angles defined based on a 3-2-1 rotation. My specific questions are:
1) is my understanding and calculation of bank angle correct? If not can someone point me to a good resource for this. I was surprised to be able to find a clear definition and formula quickly.
2) by multiplying the second and third row of $C_{w-b}$ with Euler angles we obtain a set of other two angles relative to stability frame. Do these angles have any names or specific role in aerospace dynamics/control?