Late answer, but I'm hoping this clarifies what is going on. Whenever the front tire is turned, the bike will turn in that direction. In the case of countersteering, the rider steers in the opposite direction in order to cause the bike to lean, as shown in this image. The description in the image is not quite right, as both front and rear tires will out-track from under the bike, and the parts of the bike above the roll center will in-track.
If the bike is initially vertical, then the initial counter-steering input will cause the bike to turn "outwards" while it leans inwards. 544
That same outwards force that is causing the tires to track outwards (so that the bike leans inwards) is also causing the bike's center of mass to accelerate outwards. If steering right to lean for a left turn, the bike is initially turning right until the steering is brought back to neutral and then left, and vice versa. Out tracking can be used to quickly dodge potholes, you steer away from the pothole to get the tires to clear the pothole, which also causes the bike to lean, so after clearing the pothole, you have to steer the other way to get the bike back to vertical.
The steering geometry for a typical bike locates the contact patch behind where the extended axis of steering would intercept the ground. When leaned over, the ground is pushing up at the contact patch, while gravity related downforce is exerted at axis of the tire, causing the front tire to steer inwards in response to lean angle, causing a bike to return to vertical orientation (absent other inputs).
In the original question's video, there's mention of angular momentum or gyroscopic effects on steering, but at all but very slow speeds, the gyroscopic effect at the front tire results in too slow a rate of precision that slows down the self-correcting steering torque related to trail, acting as a damper (opposing self-correction), and at high speeds (100+ mph or 160+ kph), it's enough to virtually eliminate self-correction and a bike at high speed will tend hold a lean angle rather than return to vertical.
Regardless if a turn is coordinated (steady lean) or not (lean angle changing), there is a <PERSON> third law pair of forces, the pavement exerts a centripetal force onto the tires, coexistent with outwards force exerted by the tires onto the pavement. If the turn is coordinated (steady lean) the outwards force is the same as the centrifugal reaction force related to the centripetal acceleration of bike and rider time the mass of bike and rider. 544