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I am so pleased that A Twist of the Wrist II is available now on Amazon Video! I refer students to it all the time. One of the specific items I recommend it for is the great CG illustration of how countersteering works. It's much nicer to be able to stream it instantly instead of waiting for a DVD to arrive. You know what else? If you are ever looking for something to watch, when you go to the Twist II rental page on Amazon, it suggests other movies that people who rented Twist II watched, and it is such a cool list of motorcycle movies! Here's the link, take a look: https://www.amazon.com/gp/video/detail/B089ZNVBW9/ref=atv_dp_share_cu_r

I don't claim to know this for a fact, but my first thought is this: countersteering takes the front wheel out of line with the direction of travel, creating the lean, and that creates resistance and some temporary deformation of the tire, and that if you try to push the bar quickly (but without increasing the force) the tire just sort of bounces back at you and you get a wobble instead of a real direction change, whereas a STRONGER push really turns the bars and creates a larger force at the contact patch to lean the bike over rapidly. That's my thought, but I'll try to get a more technical answer for you from the boss. The main thing we are trying to avoid is riders trying to "punch" at the bar, because that creates instability and a wobble in the steering. That is easy to test, go out and ride and try a very light (low force), controlled push on the bar and see how the bike steers. Then try a much firmer push with similar control (harder but not faster). Then try a very quick, low force stab at the bar (faster but nor harder) and see what happens. Which gives you a faster and more controlled steering result? When you try this, make sure you are going at a decent speed, over 25 mph, so that you don't accidentally oversteer and lean too far - it takes a lot less force to lean the bike over a very slow speeds so that makes the whole exercise of playing around with the steering more difficult.

Yes, the amount of lean angle will depend on how LONG you push on the bar, and the steering rate (how quickly the bike leans over) will depend on how HARD you push. And your last sentence is stating it correctly, yes.