Counter steer happens at all speeds at all times on a two-wheel vehicle, it's just hard to notice at slow speeds, and confusing because you do steer right during the stable lean angle portion of a right turn, as depicted in the first image in that Wikipedia article.

I think counter-steering is misunderstood since people assume it means that if you're turning right then your steering has to be left of center. But counter-steering is referring to the direction of change, the steering delta, not the absolute steering angle. You always steer in the opposite relative direction from what you'd do in a car. This is still true on a bicycle at 1mph.