Bad news. Trump and Hegseth do not have the authority to rename the Department of Defense, no matter what they put on a Web site. That requires an act of Congress, which hasn't happened. And probably won't, because even if they could convince Congress to do it, that would require them to
ask... and their whole modus operandi is based around pretending to have authority they don't have.
Calling it the Department of War is accepting that Trump's the King.