I wonder if the LAPD has a policy whether off-duty cops need to identify themselves if they need to take out their gun when off-duty? Just as a precautionary measure to alert everyone? I can imagine a scenario where a "good guy with a gun" thinks an off-duty cop is an active shooter and the situation becomes even worse than it already is. I would think just the simple act of stating that you are a police officer would/could make a person stop from approaching (it may not stop everyone, but it may stop some people).
There's some unconfirmed details in this LAIST article that I haven't seen published elsewhere. Like the cop's attorney says he didn't say he was an officer. The family's attorney says he did, and that the cop was still on the ground when shooting. Also, the mother supposedly was shot in the back.
LAIST: What We Know, And Don't Know, About The Fatal Costco Shooting By An Off-Duty LAPD Cop