PS. I would also highly recommend these sources which have been covered before on HN:
Curves and Surfaces by Bartosz Ciechanowski [1]
A Primer on Bézier Curves by Pomax [2]
If I understand your suggestion correctly, I believe what this buys you is directional control of your spline at the spline endpoints and a more constant parameterization (one that doesn’t accelerate or decelerate at the spline endpoints). Perhaps the most common alternative is to insert duplicate knots at the spline endpoints, inserting as many extra knots as you need to reach the support requirements of your segment (degree minus one, right?). The problem with this is that you end up with several segments decreasing in size to zero, so the parameterization decelerates to zero. The derivatives also shrink to zero, which can cause problems depending on what you’re doing. Another common technique is to pretend the control points keep going straight after they end, meaning insert implicit ‘phantom’ control points beyond the first & last control point of a spline that matches the direction & magnitude of the first & last explicit segments. (Freya discusses this at 46 minutes into the video.) If your spline was degree 2 and starts with control point p1, then you could insert the phantom point p0 where p1-p0 = p2-p1. This gives you enough control points to evaluate the spline up to your explicit endpoint p1, but you lose a little bit of control over the tangent at the endpoint. If you wanted a different tangent than p2-p1, then you could, as you suggest, add explicit phantom control points. The continuity is the same in all three cases here, but the direction and tangent magnitude are unique to each approach with your suggestion offering the most control.
l(t) = t x0 + (1-t) x1
m(t) = t^2 (x0 - x1) + t x1 + x2
I wonder if the beginning can be skipped if the first video was already seen?
It looks similar because it's a sequel to the original video.