Not absolute. In Atlantic County there is an Egg Harbor Township and an Egg Harbor City, both served by radio, seperated by 15 miles. Their station's ID's are distinctly different.
However, it DEPENDS on what the FCC typed in the license hanging in their studio! On the net, in the FCC database, it may say "Township". BUT if someone in D.C. typed "Park" on the official license certificate sent to them, that's the final word.
On the Engineering Board, there are many stories of fines being overturned because the record in D.C. does not match what the Station was sent...and the Station's on-hand paperwork is always held up as the final standard. Never lose it, and make copies, notarize and store offsite. Sometimes the FCC doesn't update their database when approving changes, especially AM.