This is incorrect. The page is somewhat unclear, but you are confusing the requirement for photo ID with the requirement for ID.
You do not need photo ID when travelling domestic. A BAEC card or company ID card is valid identification.
You do need photo ID when checking bags. A BAEC card is not photo ID, therefore does not meet this requirement.
If travelling domestic and checking bags, photo ID is required. The fact that BA staff do not enforce this requirement is another issue.
Thanks for all the comments, and will take passport next time if case has to go in the hold
