Perhaps just me, but this "push left/right" advice messed up my understanding of counter-steer when I was learning and I found myself being unable to turn effectively. I would push the handlebars down or slightly forward and be able to turn but innefectivly.
Far better in my opinion is advice to simply "turn left briefly to go right". As in actually turn the wheel left, then let the bike balance by going right. Moto control explains it effectively: https://www.youtube.com/watch?v=ZLqyN5yy6I8
Depends on your velocity. Push steering works great when riding through a corner at speed. When turning from a stop you need to momentarily turn the opposite direction to initiate the lean.
What's not explained in the text on that video is that braking while leaned over causes a motorbike to steer itself upright.
You can't go around a corner on a bicycle or motorbike at more than about 5mph without counter-steering. You have to tilt the bike into the turn or it will just fall over. You tilt by briefly steering in the opposite direction, which rolls the bike in reaction. This is counter-steering.
Braking while leaned over applies a torque to the steering column towards the side which is closer to the road, i.e. steering into the corner. This torque is due to the width of the front tyre; the contact patch is at an offset to the steering axis. This torque counter-steers the bike upright.
Braking also changes the geometry on most bikes as the forks compress and the bike dives at the front, steepening the steering angle, which makes steering more responsive, so the upright counter-steer is more pronounced.
There's also target fixation. You tend to steer where you look, and if there's an enormous hazard coming at you, you risk steering right into it. Look for the escape routes.
There's also a ground rush panic effect. When something is coming at you really quickly, your instinct is to tense up and brace for impact. When landing a parachute, you're trained to look at the horizon and depend on peripheral vision to feel the ground coming up to meet you, so you flare at the right time. On a bike, it ties in with target fixation; don't look at the fast thing coming at you, look at the escape routes.
Motorcycles and bikes are not exactly the same in this respect. You can steer a bicycle without counter steer by shifting your weight to the left and right. Its possible to go around corners without touching the handlebars like this.
The fact that you can’t really make a motorcycle move by shifting your weight is probably why it is so important to teach countersteer- so people know what do to when they need to swerve quickly. Another counterinuitive thing (for me) is that motorcycle steering gets much heavier at high speed while bicycle steering is virtually effortless at most normal bicycle speeds.
Far better in my opinion is advice to simply "turn left briefly to go right". As in actually turn the wheel left, then let the bike balance by going right. Moto control explains it effectively: https://www.youtube.com/watch?v=ZLqyN5yy6I8