Looks like your up to date.
I just tried your site with Google Chrome and the Flash customizer, and made 2 designs, one with Public checked, and one with it unchecked (uncheck the box for show in public gallery in customizer step 4).
Both items show up for me in the cart OK and I have not logged in.
Here’s the screen shots of each design in my cart, first was public, second was private.
I don’t see any issues here. Can you test with 24×12 magnets product and verify it works for you too? Maybe something specific to your product set up is breaking it for some products and not others?
Maybe check your browser security settings as well. Perhaps you are blocking browser data or cookies or something that would block you from being known as the author of the design?