This sends the X-Y plane to the X-Y plane

You're right. Rotation alone won't deal with parallel planes. I guess that's why I included:

(assuming they are not parallel)

The nice thing about rotation-only examples, as a starting point, is that they are easy to visualise, whilst getting away from thinking in terms of rotations occuring (only) around the orthogonal axis. Once you've mastered thinking in terms of rotations around arbitrary vectors, then applying a translation, once the points are in the same plane, comes fairly naturally. Reflections and sheers are somewhat harder to visualise (for me at least), but if you take simple steps, you get there.

Of course, if you're one of those that is comfortable just manipulating numbers in the abstract, then just applying someone elses formulae is all you need. But I find that without some grounding in visualising the way transforms combine it is nearly impossible to acquire a confidence in the results. And if you need to modify those formulae--to impose one or more constraints on the freedom of motion--then it is neigh impossible if you do not understand how the formulae work. YMMV.