Regarding the assignment insignia, if you take TOS as a whole then their meanings aren't really decipherable. Scotty sports the round insignia in "Where No Man Has Gone Before" and then puts it away for many episodes and then dons them again briefly in "Lights of Zetar". McCoy occasionally changes his insignia form round to the spiral. The star seems to be consistent with the yellow/green shirts.
However, as you'll find out, the delta insignia is also seen on other ships and starbases and also mixed in with the insignias that have no room for these little differentiators, like Decker's insignia from "The Doomsday Machine".
So, the shirt colors are likely for divisions but the insignias are something more personal, like a decoration rather than anything meaningful, IMHO.