I was recently in downtown LA for a convention and there was at least one pedestrian signal that had a white walk signal that showed for 1-2 seconds. If you had your head turned to the side while waiting for the walk signal, you would likely miss it before getting the red flashing "don't walk." LA is the least friendly walking city I've ever seen.
There is often a countdown timer as well, and the article implies that was so in this case. Since the timer has to be set to accommodate the slowest pedestrians, the majority of walkers will in fact be able to safely cross even when commencing after the flashing/countdown has started. A reasonable adult can certainly gauge whether the remaining time is sufficient to complete the crossing.
Note that for autos, the equivalent is the yellow lamp in a stoplight and in that case the rule is that the vehicle must have entered the intersection before the lamp turns red. So LA seems to have standards that are paradoxically more stringent for pedestrians.