According to the linked website,
The reference angle of an angle θ in standard position
is the acute angle r
formed by the terminal side of θ
and the horizontal axis.
When we are measuring the angle between the terminal side of $\theta$ and the $x$-axis, we are implicity choosing the side of the $x$-axis that it's closest to. That's why the direction changed. In the first example, the terminal side of the angle $135^\circ$ (i.e., the ray going off to the upper left) makes a $45^\circ$ angle with the negative $x$-axis. In the second example, the terminal side of the angle $240^\circ$ (i.e., the ray going off to the lower left) makes a $60^\circ$ angle with the negative $x$-axis.
Note that $-1860^\circ=-1860^\circ + 6\cdot(360^\circ) = 2160^\circ - 1860^\circ= 300^\circ$. This is in the 4th quadrant (the lower right one) and the terminal side of $300^\circ$ makes a $360^\circ-300^\circ=60^\circ$ angle with the positive $x$-axis.