We wish to construct a general rotation $\mathbf{R}$ of a coordinate system by composing three elementary rotations $\mathbf{R}_1, \mathbf{R}_2, \mathbf{R}_3$, so that a vector $\mathbf{v}$ is rotated to $\mathbf{v}'$ via $$ \mathbf{v}' = \mathbf{R} \mathbf{v}. $$ A common convention is to take the rotations about the $z,x,z$ axes in that order, such that the rotations are intrinsic; i.e., the axes of rotation are (in order) $z, x', z''$, where the primes indicate that the coordinate axis has been rotated: $$ \{x,y,z\} \overset{\mathbf{R}_1}{\to} \{x',y',z'\} \overset{\mathbf{R}_2}{\to} \{x'',y'',z''\} \overset{\mathbf{R}_3}{\to} \{x''',y''',z'''\}. $$ In this case, $$ \mathbf{R} = \mathbf{R}_3 \mathbf{R}_2 \mathbf{R}_1 $$ and the elementary rotations are given by $$ \mathbf{R}_1= \begin{bmatrix} \cos \phi & \sin \phi & 0 \\ -\sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{bmatrix} \quad \mathbf{R}_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \end{bmatrix} \quad \mathbf{R}_3= \begin{bmatrix} \cos \psi & \sin \psi & 0 \\ -\sin \psi & \cos \psi & 0 \\ 0 & 0 & 1 \end{bmatrix}. $$
The aim is to first rotate in the $xy$-plane by $\phi$, achieved via $\mathbf{R}_1$, then to rotate in the $z'y'$-plane by $\theta$, achieved via $\mathbf{R}_2$, and finally rotate in the $x''y''$-plane by $\psi$, achieved via $\mathbf{R}_3$.
Question: Why does this sequence of rotations effect an intrinsic rotation, instead of an extrinsic one? $\mathbf{R}_2$, for example, looks like it effects a rotation in the $yz$-plane, not the $y'z'$-plane. However, this example appears in two textbooks (Thornton and Marion 2004, Goldstein 1980) as an intrinsic rotation. Both texts are very clear that the sequence of rotations is intrinsic. All matrices are given explicitly as I have reproduced above.