Well, when you are actually in a constant radius turn at any speed, you are always turning the handlebars in the direction you are turning.
Countersteering is used to impart roll velocity to the bike one way or another.
So at low speed, the gyro forces are so low, that you really dont need to countersteer to roll the bike over, your body weight will do it. At high speed, you need to countersteer to roll the bike over, but once in the turn, you'll still be steering the direction you're going.
However, another caveat. If you get on that superbike, get it rolling at 5 mph, and sit bolt upright and dont move your body at all, if you steer left, the bike will tip right, and you'll immediately have to start steering right to catch the bike.
The physics behind countersteering are actually pretty straighforward and well understood (despite what that bloke on the no-bs bike page says).
A bike tyre, the same as a car tyre, generates a lateral force when it has slip angle. So if you're going along in a straight line and you turn the bars left, the front tyre will generate a leftward force, AT GROUND LEVEL.
But the centre of gravity of the bike and rider is waaaay above ground level, so this force actually generated a rotation force, but a leftward force below the cog causes the bike and rider to rotate rightwards, tipping the bike into a right hand turn.
At this point, if you just kept the bars pointing slightly left, the front wheel will carry on creating a leftward force, you'll keep tipping right, and in about 0.3 secs be skidding along the black stuff on your pants. So as the bike starts to tip over, you instinctively start to steer right (caster helps I think), which starts to generate a rightwards force, which wants to tip you leftwards, thus slowing down your rightwards tip.
And then you reach a point of equilibrium where rotation force caused by the lateral force from the tyres balances with the rotation force created by your cog being off centre, and you carve a nice turn.
But you can see from that that the further off centre your COG is, the more lateral G you can pull in a turn of a given radius, hence hanging off the bike.
(NB: I've never ridden a motorcycle, I'm just a theorist, so If anyone wants to disgree based on practical experience, I'm all ears.)