This crops up all the time.
At very low speeds, you turn the bars right if you want to turn the bike right.
Above those low speeds, you lean... yes?
No.
As proven in the videos, you turn the bars slightly left, you will lean the bike right. Many people push forward on the side they want to turn, so push right bar to turn right... right?
Now - When you go to lean, the first thing that moves will be your head. Shoulders will follow, then the body... but wait - Shoulders move? Yep, as you lean your shoulders juuuuuuuuust a fraction, you put slightly more force on to one arm than the other... and that's enough to move the bars to start the counter-steer.
OK, so now you're banking over. This is where most people think they're turning the bars in the turn direction... and they're correct, to a degree. Because you're banking, you're basically tipping the bike over on it's side, so you turn the bars back to stop this and maintain lean angle throughout the turn.
Thus a bike turn is a controlled fall, in essence. But to start the lean, you counter-steer.
On a Cruiser, it is much more noticable - The bars are wider and you have almost no weight on your feet, so you have no option but to countersteer and you really feel it in those arms. I actually pull on, or hang off the opposite bar during some turns. On my Tourer, for example, it's less noticable but still works the same way.
I'm still conscious of doing it, but it blends with everything else into an overall feel for the handling.