I don't know if anyone else will go along with this, but I have my own interpretation of the different badges worn by officers of different units in TOS. Some people interpreted the different shapes of the badges as indicative of different ships, but the officers in the O-Club on Starbase 11 in "Court Martial" who confronted Kirk were obviously not from the Enterprise, yet seemed to know him, and wore the same badge/insignia.
I interpeted this arrowhead badge as being a unit larger than just the NCC-1701 Enterprise. Maybe it represents the First Fleet, or some other conglomerated unit or mission-task-force within Starfleet. My first thought back in the 1970's is that Kirk's Enterprise was attached to a specific organization of ships that all serve the same function: a consolidated mission of combined exploration, peacekeeping, and certain kinds of research. Other ships and wings of ships might serve other functions, or combinations of functions. Commodore Decker may have been commanding the Constellation on a completely different mission, one very different than the Enterprise. Or perhaps his ship was part of the Third Fleet. And Captain Tracey's Exeter may have been on yet another mission, or perhaps part of the Second Fleet.
And so on.
Could those badges be interpeted as logos for projects/missions/wings/fleets, like the shoulder patches for Archer's NX-01 Enterprise and the earlier NX Project? If so, what do you think the pennant logos would look like, as derived from the badges?