As my first Commanding Officer when I was in the Navy had a rank (not position) of Commodore, I’ll give this one more try, and then I’ll give up. As wiki states about the U. S. Navy:
“The one star rank appeared again in 1982 with the title of Commodore Admiral. The next year, after numerous protests to the Chief of Naval Operations, Commodore Admiral was changed to simply Commodore. In 1985, Commodore had also begun to appear as a position title for senior Captains who commanded Destroyer Squadrons, Patrol Boat Flotillas, and Naval Aircraft Groups. To prevent confusion between the title of Commodore and the actual rank, the one star Navy admiral rank was changed to Rear Admiral (Lower Half).”