Eyeroll.
Adding "Department of War" as a secondary name of the department is dumb as fuck- but it did happen, and it's perfectly legal.
The Congress set its statutory name, and the executive *also* has the authority to name it however he wishes for the Federal Government.
This system is silly, and leads to dumb ass shit like the Gulf of America, but it is legal. We should probably fix it, so that we don't have to deal with this dumb shit in the future.
So while it is true that "It is called the Department of Defense", it's nothing but pure unadulterated copium to say "There is no War Department".
There is, in fact, a War Department, and it is in fact also called the Department of Defense.
Some, in fact, would argue that "Department of Defense" is the more Orwellian name for it, and has only existed for the list 60 years or so in an attempt at making US hegemony seem more benevolent.