A tricky one, but after a little research I’ve found the answer to your question.
According to Section 1803 of the California Vehicle Code, traffic law violations must be reported to the Department of Motor Vehicles within five days after a conviction is entered. The DMV then adds that violation to your driving record, and your insurance company would presumably find out about it from them.
HOWEVER, the statute goes on to list a number of violations which are not required to be reported, including, “(6) Violations for which a person was cited as a pedestrian or while operating a bicycle or a motorized scooter.”
So there you have it, the answer is no, running a red light on a bike cannot affect your driving record or auto insurance.
One thing to keep in mind: just because it’s the law, doesn’t mean everyone (including the judge) knows it. If you ever find yourself in a California court for a hearing on a traffic violation committed while riding a bike (or motorized scooter), make sure to bring a copy of this statute with you, and remind the judge/clerk that your conviction shouldn’t be reported to the DMV. Just try not to be a smug jerk about it, ok?