I have absolutely nothing to back this up, but I always thought that maybe each ship in starfleet having a different insignia might have been a nod to the American space program, which was reaching for the moon as this series was being created -- each Gemini and Apollo mission (I believe) created its own unique patch/insignia that the astronauts wore, even though they were all using essentially the same type of spacecraft.
You believe rightly, in the main. From Gemini 5 on the astronauts designed, or had designed for them, mission patches. It started as consolation for the no-names policy adapted after Grissom and Young picked ``Molly Brown'' for their capsule's name. If all this squabbling sounds remarkably petty to you that's because it kind of was.
After it turned out mission patches were very popular as public relations items and for mission team morale there were patches retrofitted onto the Mercury and the first two Gemini flights.