Close, but in my day (75-87) there was no rating for firefighter. Every single recruit went though fire-fighting training. However, there was what was called, "flying squads" for immediate attack of fires. I could have that term wrong. There were also Boatswains Mates designated for damage control who would back them up once geared up, Don't know how it works today, sure some current Navy can verify or correct me. Again, in my day no fire-fighting rating.
I was and ABF on the flight deck and we practiced all the time for flight deck fires. We had no special rating for those drills.