It's dangerous to sit out in the middle of the intersection, especially large ones.
If there is a car in front of me already in the intersection, I may also move into the intersection depending on the circumstances.
If the volume of oncoming traffic is light, I will proceed into the intersection behind the car in front of me. If there are two or three lanes of heavy oncoming traffic, I am not going to risk making my turn on a red, the chance of an oncoming car running the red is too great.
If I am the first car waiting to turn left, I will only advance into the intersection if the cars approaching perpendicular to me on my left have stopped for the red light. This reduces the chance that a red-light runner will hit the left side of my car.
The other thing to keep in mind is keeping out of the intersection allows emergency vehicles more room to get through should they need to.
My safety trumps your inability to remain patient through another light cycle.