How does the Kodokan classify this technique?
Timing v. structure (deashi-harai v. kosoto-gari)
I would guess that the 'sticker' label is applied to a throw where tori's leg/foot starts in contact with uke's leg, but I think these variations would not be classified as the same Kodokan technique. The key distinguishing feature between a kosoto-gari and a deashi-harai is whether uke has weight on the leg under attack.
From Kodokan Judo Throwing Techniques by Toshiro Daigo p. 157, this difference in uke's weight is described:
Differences between kosoto-gari and deashi-harai
At the moment uke steps his right foot forward onto the mat and transfers his body weight onto it, tori pushes him towards his right heel (or both heels) to break his balance.
The tsukuri is performed at the moment uke steps his right foot forward and begins to transfer his weight (before it is completely transferred).
My interpretation is that 4 is really a kosoto-gari or kosoto-gake. The slow-motion version at the very beginning shows tori disrupting uke's structure by moving uke's weight to the ball of the foot and toes only, which is analagous to the way the traditional kosoto-gari shifts the weight to the heel. The attack is on a leg that is weighted.
The 5 is a deashi-harai. Uke's foot is unweighted as tori lifts it, and pulling towards the rear corner forces uke to try to weight the foot in the air, and uke falls when they cannot put the foot down.
For the other versions, the distinction is less obvious. You can throw this as a deashi-harai by making contact then forcing uke to step and sweeping immediately, in which case uke's leg will not be obviously elevated before the throwing action. If forced to guess I would say deashi-harai for 1, and kosoto-gari for 2 and 3.
kosoto-gari v. kosoto-gake
I confess that I do not understand this distinction well. From Ibid, p. 162:
The opponent steps his right foot onto the mat, shifts his balance, and, at the right moment his weight falls over both heels, tori reaps from behind his right heel wit h the left foot shaped like a sickle (the toes bent inwards) toward the direction his toes are pointing, reaping by sliding the foot across the mat.
Essentially, at the moment the opponent's balance falls over his right foot, put the sole of the left foot onto the back of his heel, and scoop him up by sliding the left foot upwards, synchronizing this with the movement of both hands. These techniques should not be confused.
In an explanation of deashi-harai of Waza no Kaisetsu-- deashi-harai, Mifune, 10th dan explains the subtle differences between sweeping hooking, and reaping:
"There are three movements in ashi-waza, 'sweeping', 'hooking', and 'reaping.'
Sweeping is similar to brushing an extremely light object away.
When hooking, you execute the technique as if pulling a rooted plant out from the ground.
Reaping is similar to the movement of reaping and cutting off a plant at its root with a sickle.
So what is a sickle motion that scoops? I'm not sure, but I do not find this distinction nearly so important as whether a throw requires timing or disruption of structure.
I would classify it as a variant of kosoto-gari, kosoto-gake, or de-ashi-barai, depending on the particulars. Teaching it as "the sticker" rather than one of those looks like a valid choice.
I wouldn't worry too much about which bucket it goes in. Judo's taxonomy of techniques is a pedagogical framework with clear distinctions imposed onto the reality of a smooth continuous space. Focusing on the framework rather than the reality is looking at the finger rather than the moon. It's okay for techniques to exist outside of the classification scheme.
Shintaro Higashi seems to call it a deashibarai based on it being a kind of sweep from the outside of the foot. This is defensible. But judo throws are officially classified, technically, by the kind of motion they use, and this doesn't look like a barai to me--it's closer to a hook or gake. There's the heart of the problem: the taxonomy is really using multiple criteria, cutting across different dimensions in different contexts.
1 looks like my grand-sensei's De Ashi Harai. 4 looks like Ko Soto Gama.
In general, when uke is stepping at you and you time your attack to when the foot touches, but move it before the weight transfers onto it, they fall.
When uke is moving away from you, capture the foot and hold. If he's trying to move his foot back, why let him? You'll notice the kuzushi if you try this. I use "sticky" Ko Uchi Gari when chaining it with O Uchi Gari. I hold until uke pulls his foot out (if he's able to do so without falling) then as he's moving that foot, I move both my feet, and switch my attack to the other leg.
I really don't like to go for a foot sweep when my uke isn't moving, and most of these videos are showing that.