@Sci I doubt that Starfleet was a generic term used for the fleet, and even if it was, there are still certain overtones that come with it in the Mirror Universe for those who remember the tranny of the Empire, not to mention the very existence of Memory Omega seems to be about preventing the mistakes of the past and moving into a new future. The Commonwealth being a new space power would not want to "rock the boat" sort to speak or give impressions they were revisiting that past in any way. That would not be beneficial to their forward thinking process IMO.
I think that everything you suggested about what the Commonwealth would do is entirely plausible. I just also think the possibility that the word "Starfleet" is seen as a generic term rather than a specific term, like "Air Force" or "Navy" or "Luftwaffe," is also possible.
(After all, there was a lot of resentment of the Kingdom of Great Britain and its armed forces when the Thirteen Colonies declared independence -- yet no one suggested we should call it the United States Maritime Forces just to avoid using the same word in its name as the Royal Navy.)