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Sorry i meant to say is that how no-hands works!
Counter steering works just fine without even thinking, and can be so slight to initiate a turn, which is probably why people are not aware they were even doing it. For a quicker turn it feels like i counter steer to throw my weight quicker, and after that the brain simply rebalances everything, by which time physics of the rest of the system has combined with the rebalancing to perfectly follow the sequence you described!

Is it this?
Moving body to one side causes bike to move to other side. Tyres resist this motion but in doing so a torque is applied to front stem, so wheel turns slightly to same side as rider. Due to the already changed centre of gravity this causes/allows bike to turn in same direction as wheel just turned