Now that I think about it, you're both right.

When riding my (drop bar) bike using method (2) it feels like I'm just putting my weight directly downward on the (inside) bar - but in reality, I expect I am actually applying a turning force to the bar in the opposite direction of the turn. It doesn't feel like this because it appears to have no visible effect on the angle of the bars, which might even actually be slightly turned in the same direction as the turn when going slowly.

I've never ridden a motorbike, where it sounds like counter steering is a more deliberate action.

Thanks for the clarification!