Different states have different rules. Berberis vulgaris, also known as common barberry, is banned in most states but it is not ornamentally important so has never been widely available in the nursery trade. Berberis thunbergii, Japanese barberry, has traditionally been the one sold for garden and landscape use because it has several colorful selections. This has indeed become invasive in many places, and is specifically banned in most of them. It may often be called “common barberry” simply because it is so widespread, but it is a different species.
There are two sterile barberries in the Proven Winners ColorChoice line – Sunjoy Mini Maroon and Sunjoy Todo. Sunjoy Mini Maroon is a Berberis thunbergii so is technically banned – there is work being done to get exemptions for it and other barberries that claim to be sterile, but so far, none have been made. Sunjoy Todo is a hybrid so may eventually earn an exemption but that has not occurred yet in any state.
So after I wrote all of that: none of our barberries are B. vulgaris. They are either B. thunbergii, or in the case of Sunjoy Todo, a hybrid between B. thunbergii and something else (but definitely not B. vulgaris). Based on the link you sent above, we should, in theory, be able to send B. thunbergii to Ohio but I’m not sure if there aren’t other laws – many states are in the process of enacting bans.