Youre mistaken. A "rate" is where you are on the E1 - E9 paygrade scale. A "rating" is your assigned job, what you get after A school. A Fireman has a rate of E-1/3. He does not have a rating because he hasn't been to A school. You can also "strike" for a rating by testing into it, but thats rarer.

There is more history about this confusing system here Note that this is from a .mil site specifically about Navy history. The article is from 2019.

The United States Navy's enlisted rank and rate system is unique among the armed services. The first point of divergence is the term "rate," used in the Navy rather than the more-familiar term "rank," which is reserved for naval officers and warrant officers. The second unique aspect of Navy enlisted rates is the inextricable linkage of rates, which represent a Sailor's pay grade, and ratings, which denote an occupational specialty. For example, where a notional Sergeant Smith may have a military occupational specialty (MOS) of infantryman in the Army, he would simply be designated Sergeant Smith, both in conversation and on official documents. A Sailor of equivalent rank/rate with a rating of boatswain's mate would be Boatswain's Mate Second Class Jones. Thus, the Navy combines rates and ratings in Sailors' titles.

To complicate matters further, the Navy considers Sailors in the E-1 to E-3 pay grades "nonrated," meaning they do not yet hold a rating.