I suspect there are more symbols used in the chest insignia than the standard three we saw. For instance, Nurse Chapel (blue) has a medical cross, whereas McCoy (also blue) has the sciences oval signifying his broader life sciences duties in addition to CMO.
Official or not, I'd say there could be lots more that we haven't seen. Between division colors and rank and department symbols, one could get a broad variety of shipboard posting indicators by appearance.
Plus, we've seen Starbase Commodores in red when you'd think they'd be in command gold. So many combinations should be possible, not strictly the command star w/gold and science oval w/blue, etc.