Commodore can refer to a Rear Admiral (Lower Half), which was Grace Hopper's rank (IIRC it was awarded long after her retirement). Commodore can also refer to a group commander -- for instance our Destroyer Squadrons are commanded by a Commodore, who is an O-6. Same with Naval Special Warfare Groups.
Commodore as a title and a rank has a long and storied history all the way back to the days of sail in the British Navy.
In the UK Brigadier is still similar - “Senior Colonel” and not a general officer.
My one big regret in coming to work for what was then the Defense Communications Agency in September of 1989 was that I had missed meeting her wandering the halls of the basement of the Pentagon by just a few months. But I did work with some people who had known her personally.
I really wish I could have met her in person.