Several years ago I made a decision that I would cross streets at intersections only when the light is green for me. I decided I needed a way to slow down a bit, observe the world around me, and appreciate the fact that I do not have to be in a hurry.
I've been to New York several times since I made this decision. I am well aware that people cross when it's safe, not when the light is green for them. So when I stop at the corner, I am sure to move myself all the way out of the way of other pedestrians.
It works out fine for me, and I avoid the wrath of New Yorkers in a hurry.