When you're in motion, as per case 2, turning the handlebars has an effect on your uprightness. In fact, very minor turns of the handlebars can correct your tilt fairly easily.
In case 1, turning the handlebars will help you almost none at all.
Furthermore, if you were to lock the handlebars straight ahead, I think you would find it a lot more difficult to stay upright indefinitely.
That says to me that there is at *least* one other thing that is happening
While I expect gyroscopic effects do come into play to help you stay upright, I don't think your arguments suggest proof.
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