$\color{red}{\text{Ordinary definition}: }$Let $f(u,v)$ be the angle (note that $0\leq f\leq \pi$) between the vectors $u$ and $v$ in $\mathbb{R}^2$, uniquely determined by: $$\cos(f(u,v))=\dfrac{\langle u,v\rangle }{||u|| ||v||}.$$
$\color{blue}{\text{Oriented angle definition:}}$ Let $u = (u_1, u_2)$ be a vector in $\mathbb{R}^2$ and define $u^{I} = (-u_2,u_1)$. Then the oriented angle, $a(u,v)$ between two vectors $u$ and $v$ in $\mathbb{R^2}$ is given by:
$a(u,v) = f(u,v)$ if $\langle u^{I},v \rangle \geq0$
$a(u,v) = -f(u,v)$ if $\langle u^{I},v \rangle < 0$
Note that $a \in \ (-\pi,\pi]$
It would appear that this definition cares about whether or not $v$ has "rotated" around $u$ and whether or not that "rotation" was clockwise or not. I don't know whether that's clear/correct, though. Here are some examples:
Assigning an angle between $0$ and $\pi$ is not sufficient to provide an orientation for the angle.
Orientation provides information about the position of two objects with respect to each other. For example, if we are standing on the equator, 10 meters between us, then saying "I am 10 meters from you" is insufficient orientation information. It would be necessary to say "I am 10 meters west of you," or "-10 meters east of you" or something that. And then you would be able to say "I am 10 meters east of you." The point of reference matters.
So for angles in a plane, simply measuring the non-reflex angle between the two suffers from this problem. You really need a sign and a convention to clear things up. Typically we use the right-hand rule so that measuring in a counterclockwise direction from the first to the second is positive. Reversing the order of the characters in the pair will reverse the orientation between the two.