re: What are the rules for running red light on a continuing left?

If there is the thick solid white line on the cross street you are turning onto from your original street, you're supposed to stop at that line and wait until you have a green light. It doesn't matter if you had been travelling straight or turning onto that street.

If there is no solid white line, that space is not intended to have cars stopped there, and you should continue the turn when there is no oncoming traffic.