there is a lot of variants even on basic techniques within kyokushin. The attitude is "if you can KO with it, it is all good."
I would say this is the basic version. But I am sure other kyokushin guys would disagree. some would pull the knee up more to the side.
Oh, and freaky. Note the spinning around version. Its just that in karate you expect to hit something, and start by learning how to pull back the kick with control after the hit (to kick again quickly). Spinning around is easy, but you only get one kick doing it.
Artur Hovhannisian points out the importance of pointing the knee so the kick goes through the opponent. Midori above does it as he kicks, but neglect to point it out -showing only the point where the foot starts the snap movement
Not well explained, but note how Yuki Maeda jumps in the foot pivot. He does not kick and drag the foot into pivot, but jumps in the pivot and thereby add power to the kick.
Some would "step in" the pivot, but that is more a distance closing method, and most frown on it as it telegraphs the attack to someone with Machida "elusiveness".
It is considered a bit of a beginner method. Bar Rutten (kyokushin background) show it pretty well.