I don't think it likely at all that "Battle Group" and "Fleet" are interchangeable; rather I think that battle group means more or less what it means now: a major combat formation organized at fleet level on a semi ad-hoc basis from resources available at the fleet level. (A quick digression: the buried assumption seems to be that every system has a fleet, which I don't think is true. 3rd fleet Capella is based in Capella, but it is likely responsible for a number of systems around that system).
So Third Fleet has a whole host of units and installations under its command: Several destroyers, a number of what a 21st century navy would call “major surface combatants” (the Corvettes, basically, although my historical completist mind insists there are other types, or at least classes, that we just haven’t seen), escorts (cruisers and such) and fighter wings, plus one god-awful big logical tail of freighters, transports, stations, repair craft and a veritable ****load of other stuff needed to keep everyone fed, fueled, and supplied. All this is under the command of Petrarch, and with it he is commanded to Go Forth, Do Good, Keep the Peace, and Defended the GTA. The implementation of this is left too him, as he has four stars upon his shoulder, and answers only to god and budget committees.
So when he needs to put down a rebellion someplace, or show the flag, or just Do Stuff in general, what he does is put together a destroyer or two, mate it (or them) with a fighter wing(s), add in a some corvettes and a great many (as that term can meaningfully be applied to great huge, absurdly expensive heavily manned space faring warships; i.e. six or eight is a great many) cruisers, assign a rear- or vice-admiral to command it depending on the size, and send them off to Do That Previously Alluded Stuff. This is what I judge is most likely meant by “Battle Group”, and its probably named after the destroyer where the commanding admiral hoists his flag, so that Aquitaine and her attendants are the “Aquitaine battle group”
Well. That just revealed that I think about this stuff way to hard. I’m going to go contemplate why no one will ever love me.