Hmm ok, i've never studied under Kanai senior students, but where I do they call it sumi/sumotoshi.. Maybe cause we don't speak Japanese and have bastardized everything.... But here's a clip of the same.
Sumiotoshi as I was taught it in USAF dojos is essentially with nage and uke facing each other, nage unbalancing uke to uke's outer rear. The video link above may just be showing a more circular way of getting there.
Kanai Sensei's taiotoshi has nage and uke facing same direction with uke behind nage off to one side, more in the position you might associate with setting up for koshinage. Nage unbalances uke to uke's inner front (as noted by others in this thread, Kanai's version is a forward projection into a roll, unlike the judo version).
I agree with Chuck about terminology and "when in Rome" but I am specifially trying to address the lineage and terminology of the video originally presented. How and why Kanai Sensei came to his version of this I have no idea. But it is what it is.