No, actually it doesn't. The commentator in the video makes a few mistakes, but most people won't care. The Electrek article uses the chart from the Panasonic research paper that I've referred to on occasion:
Development of High Power and Long Life Lithium Secondary Batteries
The primary gain is not from a different cell, but from reducing the exposed SoC during cycles. That same cell that lasts 300-500 cycles at 100% Depth of Discharge (DoD) has dramatically longer cycle life when reducing the DoD.
The original Model S cells in the 60 and 85 kWh battery packs are apparently NCR18650BE's. It isn't clear at all that they are different in any appreciable way, although the basics of binning and possible electrolyte tuning may be at play. In other words, Tesla may get the best output from Panasonic while lower bins of the same cell from the same line sold through, say, cheap Chinese outlets may not have the same performance.
Note, it isn't a bad thing that Tesla uses what is essentially a retail cell. What matters is quality, quantity, value, and performance. Other automakers have been unwilling to design packs that integrate thousands of small and relatively volatile cells and so they miss out on quantity, value, and performance. Tesla leveraged the overcapacity of cylindrical cell production to achieve the lowest price/kWh for any production automotive application.