It's all true! If you look at the secondary FAQ inside the FAQ on CA's DMV site, it says that if you go to an office to do the update they'll give you the "Change of Address Certificate card (DL 43) to complete and carry with your license."
But if you do it by phone or mail, "type or write in ink your new address on a small piece of paper, sign and date it. Carry this piece of paper with your license. Do not tape or staple the change of address information to your driver license."
I know that for our DVC APs, they didn't really care that DH and I had different addresses on our licenses. WA cares even less about address changes as CA does, and they don't have go bring anything with you. I had gotten my license after moving to where we currently live, but DH's license still showed our old address. He's the owner, and neither of us changed our last names, and they didn't blink an eye at all the discrepancies.