No, they mean department of defense.
The president and his cabinet don't arbitrarily decide names. There are actual procedures to go about doing so.
Alternatively if you feel like it doesn't matter since theyre all going along with it, I feel like the DOD and Gulf of Mexico are two of the very few entities it's okay to deadname, along with Twitter.
Whether you're right or not is irrelevant if nobody listens to you because of your tone.