"So ken, when you run, you start a right hand turn by taking a step to the left?"
You'll definitely initiate it with your left foot. It may not be a step to the left, but your foot is not directly under your center of gravity. That helps you lean, even when running you're not vertical when you turn. Being on the left foot hlps initiate your lean.
Consider an exaggerated case, a hard cut to the right that a football or basketball player might take. They do this by moving their left foot out very far from under themselves and can achieve a very aggressive lean and tighter turn. So in this case, they do turn right and initiate the turn by stepping left.
You'll definitely initiate it with your left foot. It may not be a step to the left, but your foot is not directly under your center of gravity. That helps you lean, even when running you're not vertical when you turn. Being on the left foot hlps initiate your lean.
Consider an exaggerated case, a hard cut to the right that a football or basketball player might take. They do this by moving their left foot out very far from under themselves and can achieve a very aggressive lean and tighter turn. So in this case, they do turn right and initiate the turn by stepping left.