Shouldn't angular displacement be defined only on $[-\pi,\pi)$? I'm used to displacement being a measure which ignores the path taken, like so:

but when I read about angular displacement, it seems to be more like a distance than a displacement. For instance, in this NASA graphic it's given as
$$\phi=\theta_1-\theta_0,\tag{1}$$
so, for instance, if $\theta_1=\frac{3\pi}{2}$ and $\theta_0=0$, we have
\begin{align}
\phi =\frac{3\pi}{2}-0 =\frac{3\pi}{2}.
\end{align}
But the linear distance travelled is $\frac{3\pi}{2}r$ and the linear displacement is $\sqrt{2}r$, so, by analogy, shouldn't we say the angular distance traveled is $\frac{3\pi}{2}$ and the angular displacement is $-\frac{\pi}{2}$?

But since angular displacements are measured mod $2\pi$ (i.e., we care only about the "principal value"), the values $\frac{3\pi}{2}$ and $-\frac{\pi}{2}$ are equivalent/congruent.
On the other hand, the correct angular analogue of "distance travelled" is "rotation angle" or "angle rotated through", rather.
