You can minimize the danger of collision at intersections by learning to stop defensively. First, make sure your deceleration to the stop doesn't surprise the guy behind you. You already know how to identify a "stale" green light -- by watching the pedestrian signal. Typically, when there are only a few seconds left, the red "Don't Walk" signal light will flash to let the pedestrians know to get out of the street. Plan ahead by starting to slow down then -- this keeps you from making sudden stops and gives the guy behind you ample warning that you are stopping (so he doesn't hit you).
Next, pay attention to WHERE you stop. Leave space in front of your vehicle. When driving a normal passenger sedan (not a van), stop so that when you look out over your hood, you see either the painted stop line on the pavement in front of you, or if you are behind another vehicle, you can see its rear tires touching the pavement. This gives you about a car-length of space between you and that vehicle or the stop line. If you drive a van, try to stop about the same distance back -- about one car length.
Stopping with space in front of you gives you room to pull forward if another vehicle comes up behind and is about to rear-end you (you're still checking those mirrors every five to eight seconds, right?). Sometimes, all you need is three or four feet to give that rear-end guy enough space to miss you. Second, if the vehicle stopped in front of you stalls or breaks down, this leaves you room to pull around, and you won't get stuck there at the light. Third, it helps you reacquire your two-second space cushion when it's time to go. And, it helps keep you out of that intersection for a moment when the light turns green, and helps you miss the red-light runner. Don't enter intersections for the first four seconds after a light change.
When the light does change, make sure you clear the cross street (look left, right, and left AGAIN) before you enter the intersection. It is not legal to enter on the green light until the intersection is clear of traffic and pedestrians (and you do want to avoid those pesky red-light runners).
Finally, do you know WHERE you have to stop? You can stop sooner if you choose, but remember this: a signal or stop sign tells you that you HAVE to stop, the pavement markings tell you WHERE. If there is a painted stop line, stop prior to crossing it. If there is only a painted crosswalk, stop prior to crossing that. If there are NO painted markings, then your stop position is at the closest place to the intersection that you have a clear view of approaching traffic -- without entering the intersection.
Keep the shiny side up!