If the title doesn't make a difference, then there's no point to insist on it. People say "the Pentagon" as shorthand for "military leadership in Washington." Not using the shorter term wouldn't do much beyond making news articles longer.

This administration says "Department of War" because they want to project an aggressive image. I support anyone who uses the legal name "Department of Defense" in an effort to reinforce an aspirational goal for the department and to remind others that the Executive Branch shouldn't be allowed to remake the entire government at will.