well the way i've been taught it on the dirt is that you push forwards with the inside hand which initiates the counter-steer (nicely setting you up on the outside of the corner) and thus puts your body weight inside of the bike's axis such that you inherently lean into the turn. you then convert that forwards pressure of the inside hand into downwards pressure to hold the lean and press down on the outside foot to balance this.

not sure which of your categories that fits in but it sure feels good when you get it right

As I read this, all I can think of is this

The Rocket Bike – Counter Intuitive Concepts

A simple and fun experiment that can be performed with a standard bike involves the use of a rocket. The strategy is to affix a small model rocket onto a handlebar to one side. The intent is that the bike will be pushed forward without rider, and the rocket will be ignited by remote means or by a timer or delay of some sort. When a bike is pushed off in a straight running line, and the rocket ignites, a steer torque is applied to the handlebars. The astonishing result is that the bike does not steer in the direction of the steer torque, but rather in the opposite direction. This astounding fact perplexes many onlookers. In essence, the bike turns in the direction opposite to the applied steer torque. This experiment demonstrates conclusively that the bicycle’s dynamic behavior is counter-intuitive from the perspective of handlebar steering actions.

Even skilled and proficient bike riders are often puzzled by the results of this experiment. The explanation lies in the fact that the rocket creates a steer torque which, yes, momentarily turns the handlebars in the direction as pushed by the rocket. This steering action in turn causes the base of the bicycle to steer in that direction. But now the bike has been forced into a lean because the ground contact support points have shifted by steering action or offset to one side, and – a lean away from the steer direction then results. It is this reversed direction lean, combined with the front fork geometry involving head angle, rake, and trail, which overpowers the rocket force and turns the bike’s handlebars into the direction of the lean – and opposite to the direction of the applied rocket torque.

In engineering terms we say that the bicycle exhibits counter-intuitive behavior. We can mathematically say, using the language of systems theoretic principles, that the bicycle exhibits “non-minimum phase behavior.”
The significance of the bicycle’s non-minimum phase behavior is that the bicycle usually does the opposite of what the intuitive mind would and might expect. As an example, if we want to steer right using handlebar actions as opposed to leaning, we actually steer first in the opposite direction, or to the left. Accident studies show that nontrivial numbers of motorcycle deaths occur in the case of less experienced riders – as when an obstacle or danger appears, the intuitive reaction is to steer away. But, in doing so the motorcycle sets up a lean towards the obstacle and the motorcycle invariantly impacts the object – and hence a serious collision results.

The proper emergency steering action is called “counter-steering,” whereby the skilled rider initiates a turn by turning towards the hazard. This action sets up a lean away from the hazard, and thus the rider can thereafter quickly and safely turn away from the hazard as a proper lean has been established. In addition, skilled motorcyclists will combine this action with an upper torso lean away from the hazard, in that sense “intuitive,” so as to as efficiently as possible extricate themselves from the hazardous circumstance.

"Countersteering doesn't mean you're turning into the rider to the left in order to veer right."

this is absolutely the definition of countersteering (or at least one definifion). here's wikipedia. steer left to turn right.

"Does this answer your question?"

i didn't ask a question, but i think the article you referenced establishes my point.

"I guess I'm still trying to figure out what your point is..."

the point as in the original post. if you go to the descending thread (there's another thread ongoing right now) there's an admonition to learn how to countersteer. i hear this all the time, learn to countersteer, this skill, like learning how to ride a wheelie. in my opinion there is no special skill to learn here. it's a phantom extra skill that doesn't in fact exist. at least in bicycling.

"I guess I'm still trying to figure out what your point is..."

the point as in the original post. if you go to the descending thread (there's another thread ongoing right now) there's an admonition to learn how to countersteer. i hear this all the time, learn to countersteer, this skill, like learning how to ride a wheelie. in my opinion there is no special skill to learn here. it's a phantom extra skill that doesn't in fact exist. at least in bicycling.