Okay then.
The image you posted in post #3 contains what I call a 2D transformation matrix. Given some fixed point, this matrix transforms the x,y coordinates of the point to the x',y' coordinates of the same point. Rotation and transformation in 2D are a bit boring. Let's take that to three dimensional space. The z axis is unchanged by a rotation about z, so that 2d transformation matrix becomes
##\begin{bmatrix} \phantom{-}\cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}##Suppose we have two reference frames. Instead of calling them the x,y,z and x',y',z' systems, I'll denote these as reference frames A and B just to keep the labels short. Suppose there exists some vector ##\vec x##. It's just a vector, an arrow in space, with no representation attached to it yet. We can express that vector as coordinates in frame A or in frame B, but regardless of representation, it's still the same vector. Let's denote these two representations as ##\vec x_A## and ##\vec x_B##.
I'll start by denoting the transformation matrix that transforms the frame A representation of this vector to the frame B representation as ##T_{A\to B}##. Read that as "the transformation matrix from frame A to frame B". In math, ##\vec x_B = T_{A\to B} \, \vec x_A##.
Now suppose we have a third coordinate system, call it frame C, and a transformation matrix ##T_{B\to C}## that transforms from frame B to frame C: ##\vec x_C = T_{B\to C} \, \vec x_B##. This is your sequence of rotations you asked about in your first post: Start with frame A, rotate the axes to get frame B, rotate again to get frame C. The question you asked in that first post is how does one construct the transformation matrix from frame A to frame C given the transformation matrices from frame A to frame B and from frame B to frame C.
We already have ##\vec x_C = T_{B\to C} \, \vec x_B## and ##\vec x_B = T_{A\to B} \, \vec x_A##. Substituting the ##\vec x_B## in the latter into the former yields ##\vec x_C = T_{B\to C} \, (T_{A\to B} \, \vec x_A)##. Since matrix multiplication is associative, this expression can be written as ##\vec x_C = (T_{B\to C} \, T_{A\to B}) \, \vec x_A##. We want the matrix ##T_{A\to C}## such that ##\vec x_C = T_{A\to C} \, \vec x_A##. Thus ##T_{A\to C} \vec x_A = (T_{B\to C} \, T_{A\to B}) \, \vec x_A##. This has to be valid for all vectors ##\vec x_A##, which means ##T_{A\to C} = T_{B\to C} \, T_{A\to B}##.
In short, transformation matrices chain right to left. That's why your R(α)*R(β) didn't work.
Since rotation is the conjugate of transformation, rotation matrices chain left to right: ##R_{A\to C} = R_{A\to B} \, R_{B\to C}##. This is a bit more intuitive than transformation matrices since you just list the rotation matrices left to right in the order the rotations are applied. The arrows help as mnemonic for rotation.
This suggests an alternative nomenclature for transformation matrices: Instead of ##T_{A\to B}##, write ##T_{B\gets A}##: "The transformation to frame B from frame A". With this, look at how vectors transform: ##\vec x_B = T_{B\gets A} \vec x_A##. The subscript A's are right next to one another, and the subscript B's are also relatively close together. This notation also makes the chaining of transformation matrices a lot more intuitive: ##T_{C\gets A} = T_{C\gets B} \, T_{B\gets A}##. Just follow the arrows from frame A to frame C, and voila!
