It is known that the group $\text{SO}(3)$ of rotation-matrices (matrices $A$ with $\det(A)=1$) are generated from three parameters. This can be expressed by the fact, that any rotation matrix is a composition of axis rotations
$$ \begin{pmatrix} \cos(\phi)&-\sin(\phi)&0\\ \sin(\phi)&\cos(\phi)&0\\ 0&0&1\\ \end{pmatrix}, \begin{pmatrix} \cos(\phi)&0&\sin(\phi)\\ 0&1&0\\ -\sin(\phi)&0&\cos(\phi)\\ \end{pmatrix}, \begin{pmatrix} 1&0&0\\ 0&\cos(\phi)&-\sin(\phi)\\ 0&\sin(\phi)&\cos(\phi)\\ \end{pmatrix} $$
The question is: Why is the second matrix (Usually called rotation around the $y$-axis ) in almost any textbook written like this?
Related to the other two matrices, I would say that the negative $$ \begin{pmatrix} \cos(\phi)&0&-\sin(\phi)\\ 0&1&0\\ \sin(\phi)&0&\cos(\phi)\\ \end{pmatrix}, $$ is conceptual more straight forward. Any help or guidance will be appreciated.