It really doesn't matter how riders INTERPRET what is going on. The whole movement of a bike is subject to the laws of physics.
I have found that the most accurate descriptions of what is physically happening to be the most enlightening done by non bike riders (scientists)
When some one says they are just leaning the bike into a corner...they are still counter steering because there is physically no other way for the bike to take on a circular trajectory, they just don't recognise the actions that are being taken..... look what happens when you stop applying the principles of push left/ Lean left if you are in a left hand bend ---- you will straighten up (same applies for a right hand bend). Now actually turning the front wheel in the direction of the turn is something you would only do at low speed when you don't need to rely on centrifugal forces to get you round. I would not dream of trying to turn the wheel INTO a high speed turn. To make it all easier to remember, and to drop the whys and wherefores of the physics involved...just remember this useful old moto I was taught. Push Left, Lean Left Turn Left. Push right, Lean Right Turn Right...simples...