You're answering the wrong question. The question is not "how does the rider initiate a lean" which is usually through countersteering.
The actual question is "Why, when you give a bicycle a good shove, does it stay upright on its own, continually changing its direction to balance itself?"
The counter steering answer to that question is rake, the fact that the steering axis hits the ground in front of the contact patch if the front tire, so that when a bike is leaned over it self-steers into the turn (the contact patch torques the wheel around the steering axis) and countersteers itself upright.
It's well known in motorbikes that the amount of rake is what makes a bike more stable vs more responsive to steering. Some folks lower the front of the bike - tighten the triple tree lower on the forks - to increase responsiveness on otherwise more boring bikes.
Of course there's more to it all, gyroscopic effects and so on, but this is the countersteering perspective.
Reviewing this comment I didn't get all the terminology right - when I said rake, I was conflating trail, rake and steering head angle, where the latter two combine to define the amount of the former.
The actual question is "Why, when you give a bicycle a good shove, does it stay upright on its own, continually changing its direction to balance itself?"