Note: all coordinate vectors mentioned below are column vectors.
I am currently studying properties of rotation matrices using Spong and Vidyasagar's textbook "Robotic Modeling and Control." The textbook explains how rotation matrices (denoted by $R$) can be used (in $3D$) to both:
express a point $p$ with coordinates $p^0$ ($0$ is the 'global' frame) in the coordinates of a rotated frame (rotated according to $R$) ($1$ is the rotated, 'local' frame) $p^0=({}_{1}^{0}R) p^1$
rotate a point $p$ with coordinates $p^0$ in 'global' frame $0$ to a rotated point $p^0_{b}$ also in 'global' coordinates
The authors of the book also claim that intrinsic and extrinsic elementary rotations can be performed on the axes represented by a rotation matrix $Q$ by premultiplying (extrinsic rotation) $Q$ by an elementary rotation matrix and by postmultiplying (intrinsic rotation) $Q$ by an elementary rotation matrix, respectively. They make this claim (where left superscripts denote a frame of reference and left subscripts denote the coordinate axes which are being represented in the frame of reference corresponding to the left superscript) here (they use notation ${}_{1}^{0}R$ in place of $Q$ ).
However, the Wikipedia article "Davenport chained rotations" states (in final section "Proof of the conversion in the pre-multiply case") the following:
The rotation matrix of the intrinsic rotation sequence $x$-$y'$-$z'$ can be obtained by the sequential intrinsic element rotations from right to left: $R=Z''Y'X$
where the meaning of $x, y', Z'', ...$ is clarified earlier in the article.
Using the premultiplication and postmultiplication rules mentioned in the textbook, the rotation matrix of the intrinsic rotation sequence $x$-$y'$-$z''$ is $R=XY'Z''$, which does not match the sequence mentioned in the Wikipedia quote above.
Do these two sources (textbook and Wikipedia article) contain mistakes or am I incorrectly understanding the subject of intrinsic and extrinsic rotations?
According to the Wikipedia article Active and Passive Transformations, a rotation of a point around an axis by angle $\theta$ is the same as a rotation of the coordinate axes with respect to which the point was expressed by angle $-\theta$. Could the reason that the Wikipedia quote order is the reverse of the order suggested by the textbook be that the textbook assumes that only passive transformations occur (change of coordinate frames?) while the article "Davenport chained rotations" assumes that only active rotations occur?
Edit: In addition, why does the first answer to https://physics.stackexchange.com/questions/337882/mixing-intrinsic-and-extrinsic-3d-rotations also use a 'reversed' multiplication order?
Any advice or clarifications would be greatly appreciated.