Are Euler angle figures wrong?

Solution 1:

Say we apply first roll, then pitch, then yaw. (I don't know whether that's conventional, but that's not important.)

After applying, say, $5^\circ$ roll, it is time to apply pitch. We now have a choice to make: Do we apply pitch around the old, horizontal $y$-axis ("world axis"), or the new one that's $5^\circ$ slanted ("local / body axis")? It doesn't really matter which one we pick, but we have to pick one. Say we apply it to the new $y$-axis. Then we apply yaw, and to be consistent we apply it to the new $z$-axis.

Here is a little bit of magic for you: this gives the exact same end result as applying each of them in the reverse order and choosing to apply it to the old axes.

So in the end, there is no reason to labor too much over whether to use the new or the old axes. Which is to say, don't think too hard about whether you want the red lines in your figure to follow the anatomy of the plane or not. Because choosing which of the six orders to apply roll, pitch and yaw will swallow that choice anyways. So just pick one of the two. But be consistent, please. Don't use the old $y$-axis but the new $z$-axis.