Just to say it, "rolling on a point" or "rolling on a heading" are fine, but while a lot of pilots call it a Dutch roll, that's a bit confusing because Dutch roll is also the term for a particular kind of extremely uncomfortable tail-wagging behavior that large aircraft with swept wings can get into.

"Rolling on a point" is basically a de-coordinated turn -- you're constantly applying the opposite rudder from what you'd want in order to turn normally.

If you roll left, the lift from the wings is trying to make you yaw left, but you're simultaneously applying right rudder, so you're yawing right. If it's done correctly, you counterbalance the two forces and your nose stays right where it is while the plane rolls around it.

It's a bit of a complex maneuver because the timing of all the inputs, but this page explains it pretty well:

[...]

3. Roll left using left aileron and rudder.
4. Watch the sight picture. Before the nose starts to move left, release the left rudder and use the right rudder to hold the nose on point.
5. At approximately twenty degrees of bank, smoothly move from left to right aileron. Do not extend the pause between the rolling moments.
6. Right aileron requires right rudder. But wait—you already have right rudder! Watch the sight picture. Before the nose starts to move right, release the right rudder and use left rudder to hold the nose on point.
7. At approximately twenty degrees of bank, smoothly move from right to left aileron. Again, do not extend the pause between the rolling moments.
8. Left aileron requires left rudder. But by now, you already know that you have left rudder. Watch the sight picture. Before the nose starts to move left, release your left rudder, use your right rudder to hold the nose on point, and so it goes—left, right, left, right, and so on.

No, rolling does not turn the airplane

You say...

"the horizontal component of lift turns the airplane"

This is not true. The horizontal lift component changes the velocity vector ‒ it turns the flight path, not the plane. The horizontal lift component pushes the plane sideways, it does not actually turn the direction of the plane, as it does not impart any yaw-wise torque on the plane.

So why do planes commonly turn when banking?

Because that sideways motion ‒ in relation to the direction of the airframe ‒ makes the plane fly "dirty", with a β (yaw angle) that is not 0. And since the common layout of a plane is such that it has lateral stability by design, it flies like a feather-cock, always turning the nose in the direction of travel. This means that the β-angle does create a yaw-wise torque. And that torque is what turns the plane.

So it is not the lateral component of lift vector that turns the plane, but the plane's tendency to align itself with the flight path. And since you have turned the flight path, this makes the plane follow.

...in the common case. This exercise is an uncommon case.

Because just as you as you can manually force a β that is not 0 with the rudder, so you can maintain the induced beta that the banking causes. That then keeps the plane pointing in the same direction, even when you are, in effect, "side-stepping".

And if you listen in the video, the instructor is indeed saying that they are "using coordinated aileron and rudder inputs". The trick here is that they use opposite rudder input compared to making a common turn. Here they are opposing the feather-cock effect instead of helping it, as would be the case for a clean turn.