Suppose $R(w,\theta)$ represents rotation about the unit vector $w$ by an angle $\theta$. Then, I am told that we can specify any rotation in 3D by a rotation by and angle $\phi$ about the $z$ axis, $\theta$ about the new $x$ axis, and then by $\psi$ about the new $z$ axis.
The notes I'm following assert that matrix for this rotation should be
$R(e_3, \psi) R(e_1,\theta)R(e_3,\phi)$. But, how are the rotation in the middle rotations about the NEW $x$ and $z$ axes. Shouldn't $R(e_1,\theta)$ represent rotation about the original $x$ axis no matter what we do?
If your angles are defined about the $\bf current$ axes, then the rotation matrix is a concatenation (as you've written above) of the individual rotation matrices about the universally accepted convention of "Yaw", "Pitch" and "Roll" axes, in that order (which is non-commutative)
So, the short answer to your question is "No", unless you specifically create a set of rotations about the original axes.