Instead of thinking steering-counter-steering, think about the underlying physics: centrifugal force - the force AWAY from the center of turn – at what speed does it start? It starts as soon as you start moving and change direction, no matter how slow; you don’t perceive it until speed and centrifugal force reach the human perception threshold.
When you travel in a straight line and turn the front wheel in one direction, your body/mass (inertia) wants to keep going straight, so you end up leaning to the opposite direction due to centrifugal force. So:
1. Once moving, no matter how slowly, turning the front wheel left will results in leaning to the right due to centrifugal force.
2. The faster you move, the harder you turn the wheel to the left, the more centrifugal force will cause you to lean to the right.
3. If you keep the front wheel pointed to the left, you’ll fall over to the right, so you need to allow (not steer) the front wheel to move to the right to follow the direction of lean to the right, thus turning to the right.
Nothing new so far, but let’s go one step further…
4. Even people believing in counter-steering think there are 2 steps in steering: step 1: counter-steer to lean, step 2: direct steer after leaning; this is the apparent experience when you do a hard U-turn – initiate the lean with counter-steer the front wheel, then direct-steer by cranking the front wheel in the same direction of the turn to full lock. But step 2 is still counter-steering when you think thru the point of view of centrifugal force.
5. In a turn, and when coming to a stop, gravity wants to pull you down, so you’d quickly steer the front wheel in the direction of the lean/fall to avoid falling. Why does that work? Because to stop falling to the right, you need more centrifugal force to the left to counteract gravity -> by turning the front wheel hard to the right (same direction of the lean/fall), the centrifugal acts in the opposite direction (left) to stop you from falling to the right. So the “second step”, turning the wheel in the same direction of the lean is to increase centrifugal force in the opposite direction of the lean, i.e., steer in one direction to increase centrifugal force in the opposite direction – the physics is the same for both step-1 and 2, i.e., there’s no such a thing as direct steering.
TLDR: Once moving, centrifugal force always acts in the opposite direction from the direction the front wheel is turned - that’s all counter-steering is about.