Sounds like you're well on the way now and have at least seen the first glimmer of the light. Keep practising.

Another thing that may help with how you handle the second part of a turn once you've used the initial push to lean the bike....

Lets say you're riding along fairly slowly. You start to fall. What do you do to catch yourself? The right answer is that you turn the front wheel into the fall to catch yourself and get the bike back under you. Or if you only turn into the fall enough to stop the fall but not to lift it back up then you end up balancing the bike in a lean and carve around a turn. This is basically the second part of the turn.



cerewa is very correct in that at low speeds or if you want to lean into higher speed turns very slowly that the amount of initial counter steering can be hardly anything at all. You can snap into a turn with a lot of push pressure or you can barely push at all and at higher riding speeds the bike will do this really long lazy slow lean into the turn. Soon doing all this and playing with sharp snappy turns and long slow transitions will be part of the fun.