In the Lecture 32: Polar Coordinates,professor traces the circle counterclockwise, but traces the ellipse clockwise.
"Which was this one here. And first we noted that this does parameterize, as we say, the circle. That satisfies the equation for the circle. And it's traced counterclockwise."
"So this is what happens at $ t = 0 $. This is where we are at $ t = \frac{\pi}{2} $. And it continues all the way around, etc. To the rest of the ellipse. This is the direction. So this one happens to be clockwise."
What is the principle behind choosing the direction of tracing the curves?
For one curve/surface, we can parametrize it differently. For example, consider an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. It can be parametrized as $\langle a\cos(t),b\sin(t)\rangle$, or as $\langle a\cos(t),-b\sin(t)\rangle$. You can verify that the first parametrization is counter-clockwise, and the second one is clockwise.
There is no principle, per se, in choosing parametrization as long as it is correct. However, sometimes we may want to choose a specific type of parametrization for various reasons. One simple example is the arc-length parametrization. If you parametrize a curve using an arc-length parametrization, the speed (the norm of velocity) will always equal to $1$, which makes life a bit easier in some cases. Another example is that, in order to use some theorems (such as Stoke's Theorem), the theorems require that the curve is parametrized in counter-clockwise direction.