It's not considered running the red light when the light changes while one has already pulled out past the stop line or crosswalk into the intersection waiting to make a left turn. This means the first two people in line can make a Los Angeles left with a clear conscience.
Two things to watch out for, though: often, the third, fourth, and even fifth driver in line to turn left will also go, using the principle that if their bumpers are close enough to each other, it's like everyone's part of the same long freight train. Then there's the person coming from the other direction, who is apparently late to make a pitch to Steven Spielberg or something and runs the red light going straight.
There are green arrows at various intersections in L.A., but since they're not marked in the Thomas Guide, it's a pleasant surprise to run across them on an unfamiliar route. But since some of them only operate during certain hours of the day, it can also be a nasty surprise to be sitting in a left turn lane, looking at a traffic signal with five possible indications, your foot easing off the brake, preparing to make the turn as soon as you make sure everyone who's going to run the red light has done so...and then getting the green ball instead of the arrow.