The equation is always
$$ \boxed{\; \boldsymbol{v} = \boldsymbol{\omega} \times \boldsymbol{r} \; } $$
where $\boldsymbol{\omega} = \pmatrix{\omega_x \\ \omega_y \\ \omega_z } $ and $\boldsymbol{r} = \pmatrix{ r \cos \phi \sin \theta \\ r \sin \phi \sin \theta \\ r \cos \theta} $
In your first case $\omega_z \neq 0$ while others are 0, and in the second case $\omega_x \neq 0$.
$$\boldsymbol{v} = \begin{bmatrix} 0 & \cos \phi & -r \sin \phi \sin \theta \\ -r \cos \theta & 0 & r \cos \phi \sin \theta \\ r \sin \phi \sin \theta & -r \cos \phi \sin \theta & 0 \end{bmatrix} \pmatrix {\omega_x \\ \omega_y \\ \omega_z } $$$$\boldsymbol{v} = \begin{bmatrix} 0 & r\cos \theta & -r \sin \phi \sin \theta \\ -r \cos \theta & 0 & r \cos \phi \sin \theta \\ r \sin \phi \sin \theta & -r \cos \phi \sin \theta & 0 \end{bmatrix} \pmatrix {\omega_x \\ \omega_y \\ \omega_z } $$
If rotation is about the x-axis, then use the first column of the matrix above and multiply it with $\omega_x$.
