If you subscribe to the notion that the different badges indicate a unit assignment, such as "First Fleet" as opposed to "Seventh Fleet", then Captain Tracey and the Starship Exeter were part of a larger unit.
Let's assume, for sake of argument, that Kirk's arrowhead badge indicates that the Enterprise is part of the First Fleet. The classmates he encountered in the Starbase 11 Officer's Club who also wore the same insignia, but obviously weren't assigned to the Enterprise, would also be attached to the First Fleet, just not the same ship as Kirk. As for the commanders and other personnel attached to other Federation starships-of-the-line in TOS, each badge would indicate a different flotilla. Decker's badge could be "Sixth Fleet", for instance.
Let's assume, for sake of argument, that Kirk's arrowhead badge indicates that the Enterprise is part of the First Fleet. The classmates he encountered in the Starbase 11 Officer's Club who also wore the same insignia, but obviously weren't assigned to the Enterprise, would also be attached to the First Fleet, just not the same ship as Kirk. As for the commanders and other personnel attached to other Federation starships-of-the-line in TOS, each badge would indicate a different flotilla. Decker's badge could be "Sixth Fleet", for instance.