Those instructions for HIIT are okay, but many of the claims they are making are unfounded. One does not need to keep moving during rest periods to remove wastes from the system. Given the extent of warmup, rest periods, and 'recovery under stress' principle they advocate, what they have is less of an HIIT workout than an aerobic interval workout, and the complexity of cycling the lengths of work and rest are probably not very relevant.
If you want to challenge your anaerobic endurance more, you need to take complete rest between intervals and make the periods long enough to let your heartrate drop down and short-term energy systems to recharge. Otherwise, you compound fatigue and end up with much less intense intervals and hence less anaerobic/high intensity benefits. I also think the same goes for using the same exercise for each consecutive interval. By always alternating/rotating between at least 2 different activities, you can minimize the extent to which local muscle fatigue will detract from overall intensity.
There is no reason why a couple minutes of dynamic limb movement a couple of half-intensity intervals won't serve as an adequate warmup, and the same for a cool down. I think doing a full 15-20 minutes of continuous aerobics beforehand is a mistake that lessens the value of the HIIT work. Better to do the continuous aerobics during a different session, or after the intervals.