Not fan of the term "countersteering". If you changed the mechanical direction that the handlebars rotated the front wheel I doubt anyone would bother to call it "prosteering".
A better way to think about this is that bikes turn by banking. The method of controlling the angle of bank does not map in a simple way to the direction of turn.
> The method of controlling the angle of bank does not map in a simple way to the direction of turn.
Sure it does! (Unless by "simple" you're ruling out derivatives). If your wheel angle is moving right, then your bank angle is increasing left (or IOW, your left turn is becoming sharper, left turn radius is decreasing). While your wheel angle is moving left, your bank angle is increasing right. Whenever you stop moving the wheel, your bank angle will stop changing and you will be in a constant radius turn.
> I doubt anyone would bother to call it "prosteering".
Not sure I understand this objection... there's no rule about symmetric or consistent prefixes. Lots of things called X have a counter-X (see also anti-X). Examples: counter-intuitive, counter-insurgency, counter-strike, counter-reformation, counter-current, etc.
Counter-steering is differentiating from steering. But there's no need to differentiate normal steering from steering, so we don't normally prefix things with "pro", unless you're talking about someone who advocates for something. I think there are a few counter-examples to that in physics though. :)
In motorcycle safety, it's rather important to call out the counter-intuitiveness of counter-steering. People's reflexes tend to do the wrong thing in tight situations where quick steering is needed. If you need to turn right very quickly and haven't learned counter-steering, you can injure yourself by trying to steer right, and many people have.
It's pretty common for riders who haven't learned counter-steering to believe that leaning hard will be sufficient, and/or that body lean is as or more important than steering. But to turn very quickly, knowing about and being able to counter-steer is incredibly important.
Well the aviation people, who also control vehicles that are controlled by banking, don't bother to come up with a special name. When teaching people to fly they first teach how the controls can be used to control the attitude of the vehicle (roll, pitch, yaw are the terms that are used). Then they start a whole new discussion of how various attitudes will affect where the vehicle will end up.
Some things can't really be simplified (and perhaps shouldn't be).
A better way to think about this is that bikes turn by banking. The method of controlling the angle of bank does not map in a simple way to the direction of turn.