I don’t know if this makes it better or worse but it never stopped being the “Department of Defense.”

The “Department Of War” is essentially stylistic preference in the executive branch. The “Defense” name is still the statutory one.

EDIT: See below; this is invalid.

I’ll add that I – giving Mike Johnson the least benefit of the doubt I could – tried pretty hard to find any instance of him referring to the DoD as the “Department of War” following Trump’s executive order; I could not.