Does motorcycle counter-steering work by precessional motion? When a motorcycle is going over a particular speed  ($\approx 20$ m/h or 30 km/h) you learn in motorcycle class the best way to steer is using the so-called "counter steering";

I can't find reliable information about this, but my understanding is that since the wheel have a very high angular momentum, the torque induced by pushing the handlebar will cause the front wheels to rotate in the specific direction.
Say we want to go left(red on the image), you would take the left handlebar and push the handlebar (to turn the handlebar right for a split second). This would then give a precessional response given by the cross product of the torque and angular momentum, causing the wheels to tilt to the left.
I might be completely wrong but I would love to get some input on this,
 A: A precessing effect might be there, but it isn't necessary.
In order to negotiate a turn you need to initiate the lean first. To obtain a lean to the right you need to steer the contact patch of the front wheel slightly to the left. In the case of a moving motorcycle: moving the contact patch of the front wheel away from underneath the center of gravity is sufficient to obtain the required lean.
As the motorcycle flows to the required lean you follow through by steering into the corner.
With a motorcycle with proper steering geometry the steering is fluid. When a motorcycle leans the front wheel will tend to turn towards the lean anyway, independent of current velocity. Some precessing effect may be involved in the overall process, but since the overall process is fluid there is no way to recognize it separate from other steering dynamics.


Incidentally, it is a different matter in the case of a setup where a gyroscopic effect does have opportunity to obstruct steering.
An example of where that happens is the case of a gyro car with a single gyro wheel:
https://en.wikipedia.org/wiki/Gyrocar
The gyro wheel of a gyroscopically stabilized vehicle is an active unit. With an active unit the gyro wheel is suspended in such a way that it can be reoriented relative to the vehicle. Strong actuators reorient the gyro wheel in response to input from a lean detection system, counteracting the developing lean.
In the luxury yacht market there are systems for gyroscopic stabilization. Those unit have very powerful actuators, strong enough to reorient a fast spinning flywheel. (To find information about that search with terms such as 'maritime gyro stabilization'.)
These luxury yacht stabilizing systems have two gyro wheels, spinning in opposite direction and when actuated these two flywheels are actuated in opposite direction. (That double gyro wheel setup allows the system to act in a way such that unwanted effects drop away against each other, while desired effect add up.)
The early gyro car depicted in the wikipedia article had a single gyro wheel setup.
A gyro car is designed to balance such that an orientation opposite to gravitational acceleration is actively maintained.
So: you would expect the gyro car to negotiate corners similar to the way a motorcycle negotiates a corner: by leaning into the corner.
In the case of a motorcycle: note that during the cornering, leaning into the corner, the motorcycle is oriented opposite to the direction of gravitational acceleration. (Inertial mass is equivalent to gravitational mass, so while in the process of negotiating a corner the inertial effect of cornering is indistinguishable from gravitational force.)
My understanding is: the gyro car depicted in the Wikipedia article (the Shilovski car) had only a single gyro wheel. My understanding is: cornering in one direction was doable, but cornering in the other direction was problematic because the response of the active gyro setup was counteracting that move.
(Using a double gyrowheel setup would have avoided that problem, but that would have made the engineering much, much more difficult.)

In the case of a motorcycle with proper steering geometry all the dynamic effects flow smoothly into each other; you don't get into a situation where gyroscopic effect is interfering with fluid steering.
Some gyroscopic effect may be involved, but you don't get to recognize it separately.
