That is a terrible explanation for counter steering. You can counter steer by leaning the body of the bike, but that is a horrible and possibly dangerous way to counter steer. To properly counter steer you turn the front wheel in the opposite direction of where you want to go. This alters the the angular momentum of wheel, and the angular inertia will tilt the bike in order to conserve the angular momentum. You do not need to tilt the bike yourself. Then, to exit the counter steer (or to straighten the bike), you point the front wheel forward again.
The name counter steering comes from the fact that you are steering in the opposite direction that you want to turn. A rider that manually tilts their bike, might keep the front wheel straight or even worse, they might even steer the wheel into the direction that they want to go. This incorrect steering is why your explanation on how to counter steer is physically dangerous.
You may have noticed that professional riders lean into the turn by quite a bit. That is not counter steering, and they are not tilting the bike over. As long as they are not going in a straight line, they are doing the opposite of tilting the bike; one might even be able to argue that they are counter tilting the bike. Those riders are not just leaning over, they are shifting their entire body over. Their butt-cracks are no longer over the center of their seats. By moving to the side (or hanging off of the side), they are altering the bike’s center of gravity, causing it to tilt in the opposite direction in order to maintain its line of movement (or keep its balance). This reduces the amount of tilt needed for turns or it can be used to get a sharper turn with the same amount of tilt. The center of the wheel has more contact area with the road, but you are not using center of the wheel when the bike is tilted over. The reduction in contact reduces the maximum speed at which you can maintain traction with the road. By reducing the bike’s tilt, they can get contact that is closer to the center of the wheel. Again, this leaning is not counter steering, but it is something that is done along with counter steering. You can counter steering without leaning over the edge of the bike. You can also lean over the edge of the bike without turning, but this will cause the bike to tilt in the opposite direction in which you are leaning in order to stay balanced and keep going straight (it look really dumb).