IIRC, the only canon TOS badges were:
1: the delta-shield used by the Enterprise crew, plus others whose affiliation was apparently not the Enterprise in
"Court Martial" and
"The Menagerie, Part II".
2: Commodores
Wesley,
Stocker,
Barstow,
Stone and
Mendez, as well as Admirals
Komack,
Fitzgerald and
Westervliet wore the starburst-style insignia. (As did Starfleet prosecutor
Areel Shaw and
Miss Piper.)
3:
Commodore Decker wore the pretzel-shaped insignia in "The Doomsday Machine"
4:
Captain Tracey wore the upright rectangular insignia in "The Omega Glory"
5: There were various other insignias and/or patches worn by other guest cast, representing other units. Among these, the most spectacular was probably
the "hand" insignia worn by penal colony personnel during "Dagger of the Mind" and "Whom Gods Destroy".
There was never a canon reference to a specific insignia for the Intrepid. Any insignia that has been associated with the Intrepid was merely non-canon fan conjecture. Any assertion that insignia are designated for specific space vessels is also non-canon, supported by the producers apparently stated intent, but contradicted by on-screen evidence. (See #1, above.)
If fans are going to speculate, then speculate we shall: I say the famous delta-shield indicated that the Enterprise, and possibly other vessels, were part of the First Fleet or some wing therein. Other insignia would therefore belong to other fleets or wings therein, and the starburst would be Starfleet Command flag hierarchy.