Originally, starship personnel (in contrast to starbase personnel) had the Enterprise's
Then, all Commodores and Admirals and Starbase personnel wore that "flower" insignia.
But then, it also became the insignia of the USS Lexington
(that's the only starship we've seen that has this insignia).
Notice Commodore Wesley's high command chair (neither Exeter
had this chair), looks like the Lexington
might be Starfleet's flagship (in the mirror universe, ISS Enterprise
apparently is the flagship there)