First of all, if you are rolling the knee and shifting your weight forward, you're torsionally flexing the board. Any board can be flexed along its longitudinal axis, some boards are just stiffer than others, and therefore less easy to see/feel.
Second, torsionally flexing the board results in a disengagement of the uphill edge, holding that will result in a turn (generally taking as long as a count of 3 from a standstill.) So I'm not sure what you meant when you said
"I don't think any of my boards have enough torsional flex to even do that (I tried it a few times as a buddy mentioned it) but I end up just turning." (That's the idea!!)
The skidded or carved turn comes from how much pressure you apply during the shaping of the turn. A skidded will result when you roll your knee, shift weight slightly forward, and hold/stay centered while continuing the pressure on the front foot. Carved turns happen when you shift your weight slighty back or pressure your rear foot after the halfway point of the turn, thus applying pressure along the entire sidecut throughout the arc of the turn.
The other method of carving (shown in the video) requires more of a "banking" motion. This allows you to essentially ride the sidecut (rail), but also makes it more difficult to change the radius of the turn (or switch edges quickly without an extreme movement to bring the board back under you and onto the new edge). The way the was demonstrated to get the shorter radius was essentially a racing style of carved turn, where the knees follow eachother and the board is tipped from edge to edge, applying a sudden pressure to carve the board out from under the body and then releasing the pressure to "spring" the board back under the body and onto the new edge. This can be done with speed on groomers (as seen in the film) but will break down when on crud/bumps/off-piste. This is because when both feet are pressured together and act as one (with the knees following eachother) they are unable to act independantly to absorb rough terrain.
Anyway, to acheive the carves in the film, lower your center of mass and actively push your board out from under you and then pull it back under you, pushing it out in the other direction immediately. All the while, keep your body quiet and your front shoulder pointed down the fall line, although it'll require your shoulders to open a bit to acheive the knee-following necessary for the move. (Picture-head on-your upper body as a clock and board as the pendulum; the smaller the clock is in height, the faster the pendulum swings because of the shorter rod connecting the two.) Again, the key to that style is pushing the board out from underneath you, not so much of a torsional move but a tipping one with alot of pressure (although torsion is still technically happening between the turns, it is not really what you're using to engage the new edge and release the old one.)
As far as pushing the tail in/out, allow your center of mass to naturally move back as the board comes around in the turn. Best way to practice this is to do a heelside turn. Drop the leading heel, rotate the knee out and shift your weight in a slight diagonal movement down the fall line (like you described), as the turn comes around, let off the pressure, balance centered over your new edge and rotate your back knee out. This reinforces the centered/athletic stance, and it also pressures the tail of the board without needing a shift aft (which you can add in depending on the radius and speed-as AAA said) and it puts you in the perfect position to start your next turn without any extreme movements. On your toeside, practice by looking back the way you came as you finish the turn and completely relaxing your ankles.
Hope the essay isn't too much. I'm off to the slopes.