I have never been a fan of the faceplant for two reasons: 1) it relies on a trained uke to keep their legs under their center and pop up, which to me is highly unnatural and 2) frankly if I faceplant somebody the last thing I want is to get them even partway up again; I'm going for an immobilizing pin. When I was training at a dojo that used that as their ura version of iriminage, as uke I never could learn the preferred response. Nage would drive me down and forward, and it was Hello, Mat. They would either start dragging me back or or want me to start gathering myself up, neither of which I found very sensible.
I like either the short, direct form, or a less vertical form of the turning version - which is called kokyunage in dojo with Tohei Sensei lineage - where uke is essentially directed forward and around nage (the initial unbalancing is not pulling uke but drawing uke ever so slightly forward onto his front foot so the circle is forward-moving), uke describing a larger circle around nage's spiraling smaller circle, and it is the "opening door" at the outside hip that allows uke's hips to enter but not his upper body.
My preferred hand position is gently attached to uke's head; I find at shoulder is is too easy to interfere with my own movement by inadvertendly pushing down on uke - eliciting resistance - and at neck it tends to make uke tense and resistive.

I don't know if you mean to use the word "push" there, but that's not how I would describe what Tissier is doing and it is not going to work. So that would be your problem.

If you make a smooth enough entry to uke's rear, without disrputing their attack too much, you can *tip them over* towards the corner where they have no leg. Then you drop your own weight and turn, and they come along with you. It is actually really easy to just dump them right down onto the mat - the difficult part is keeping control of them as they come around - you don't want them to turn INTO you and tackle, and you don't want them to stand back up before they are right where you want them.

Anyway, if it helps

1) make a clean entry and get behind uke
2) tip uke just a bit, she should be connected and off-balance
3) move yourself

I couldn't get the clip to play, but I think I get what you mean by the various styles. I have practiced all of them at one point or another. Personally, I prefer to keep it simple and not to do the big down and up movement. However, since you want to learn it, I think Cliff hit the nail on the head. You should never push your uke. Get in the same position that you would for your regular irimi-nage (i.e. all the way behind your uke, not slightly in front where they can elbow you) and rather than propelling them forward, try to take them backwards by stepping out and dropping straight down. I'll repeat that: Don't do a circular movement, just step out and drop down. If your uke ends up on the mat and can't get up again, that's their problem. If they are able to receive it, then they will circle around and it will look like you have done a circular movement. This is why Janet was saying that the technique is dependent on uke maintaining their legs under them. An unpracticed uke will end up on the mat every time, and being able to receive this technique is an art in and of itself.