The main advantage to knowing is that you can stop wasting energy with footpeg weighting or unnecessary leaning and use bar pressure. (weighting and body position have their uses but for most riders in most situations countersteering is the most efficient technique.)
That's when they call it push steering. But it's really the same thing. When you push, you are causing your bike to counter steer. You are literally pushing the bar forward, even if you think you are pushing downward, which is by definition a counter steer.
I think a lot of people assume those are different things. But they aren't. You simply can't turn at speed without counter steering, regardless of how you visualize the mechanics.
You are spot on for people who think pushing with their foot on the inner foot peg has an effect. I'm sure it does, but it's a lot of wasted effort given that the end result you are looking for is the bar turning the opposite direction, and you'll be doing that whether you are conscious of it or not. It takes far less effort to simply not think about it, and do what comes natural since your arms and hands will inevitably do the right thing without any "different" theories interfering.
It's also why 3 wheel bikes are notorious for throwing the rider high-side if they corner too quickly. It stops you from leaning, so there is no ability to counter steer. You end up behaving more like a London double-decker bus in the turn.
