Free bag is for the primary cardholder's personal travel, and up to 8 companions on an identical itinerary. Sorry to break the bad news. I think this requirement is the first sentence in the long terms and conditions for the card advertisement.

This can be more confusing since other airlines have different requirements. For example, you must pay for the ticket with their card on United, but you don't have to pay for your Delta ticket using the AMEX.

For Delta, all that's needed for the bag waiver is to have the SkyMiles number that's printed on your credit card to be in your reservation. If you have additional questions, I think there might be some other threads here with thorough details, or I can answer them here.