I have a problem with any statement that refers to Nyquist and to 'perfect' reconstruction. The reconstruction is never, ever perfect. There are always errors such as mathematical truncation (you can't compute everything to an infinite number of decimal places...), quantization (8-bits has its limitations), noise, input amplifier distortion and group delay, width of signal memory and the sinc function as applied, and probably more things. The more samples you have, the less the reconstruction depends on any one sample or calculation so it becomes easier and easier to get a good (again--not perfect, ever!) result.
As a practical matter, reconstruction from 2.01 samples per cycle would only be practical for on-paper mathematical examples calculated to many decimal places, whereas reconstruction from 100 samples per cycle will work no matter how bad the method--even just showing the dots would obviously work fine. In between those two extremes you have practical limits. With ideal conditions 2.5X is possible, but even a little bit of noise will make the results distorted and jumpy. 4-5X typically gives good results under most conditions, although again it is never perfectly correct. I don't know exactly what caused the result in the video that you mention, but I think if he had done continuous acquisition instead of a single shot you'd see a noisy, jumpy signal.