Is there a cannon reason why DS9 did not qualify as a potential Star Base, which would have made it "Star Base ___"? I know it wasn't set up as a repair/refit/resupply facility, but, in partnership with the Bajorans - who asked the UFP to come in the first place - I could easily see how it would have been valued and welcomed as such a facility. Also, it would have potentially been visited by more Star Ships, and would have had the reputation as a Port visited by powerful Federation ships, and had the "protection" if lacked as a "Deep Space" station.
Well technically speaking it was a Starbase, it's full title was Starbase Deep Space Nine.
Mentioned in the episode "Prophet Motive"
SISKO: The Federation Medical Council has announced the nominees for this year's Carrington Award. And the nominees are, Doctor April Wade of the University of Nairobi, Healer Senva of the Vulcan Medical Institute, Doctor Henri Roget of the Central Hospital of Altair Four, Chirurgeon Ghee P'Trell of Andoria, and Doctor Julian Bashir, Chief Medical Officer of Starbase Deep Space Nine. Good luck.