Commodore Decker was the only person we saw wearing the pretzel badge aboard the Constellation, as there were no other Constellation survivors to appear in uniform. There was no onscreen explanation for the pretzel badge, and Justman didn't mention it in his memo complaining about the Exeter badge. Additionally, in "Court-Martial," Kirk's former classmates giving him grief over the loss of Ben Finney wore the standard Starfleet delta, not the Starbase/Starfleet Command flower. Presumably, they were assigned to other ships and not under Kirk's command on the Enterprise, otherwise they'd be more circumspect in mouthing off to Kirk in the bar.
In "The Tholian Web," the badges were either missing or were the regular Starfleet delta.
In the seventies, a fan myth took hold that each ship had its own badge, and when those fans went to work on TNG and later productions, they carried that assumption with them when they wrote things like the Encyclopedias and Chronologies and Technical Manuals and other "reference fiction." The Defiant didn't actually get its own badge until 2005's "In A Mirror Darkly."