Back in the old Navy (1965-95) Officers were addressed by their Rank, altho Lt. Commanders were usually called Commander.
Enlisted people were almost always called by their last name, (or nickname among their division). In a slightly more formal setting, they were addressed as "Seaman Jones" or "Petty Officer Jones." The Bosuns were almost always called "Boats" Gunners Mates were almost always "Gunner."
Once you put on Khaki, you were Chief, Senior Chief (or Senior) and Master Chief. In the Chief's Mess (called the Goat Locker) Chiefs would use first names or nicknames but in public is would always be Chief, Senior or Master Chief.
Now, if you were talking about a guy in another division, you might refer to him as "that first class ET" but if you knew his name you always referred to him (or her) by name. Within a division, most guys were the same rating.
The only time your actual Rating and Rank came into play was on citations, or in the Plan of the DayDuty Rosters.
But to get to the crux of the discussion, in listing a person's formal Navy Rank/Rate, you would use the correct Rating - CTRCS,CTTC, CTR1, BM2, QM3, BTSN, etc.If they changed Rate during a career, you would list their early service as whatever, make a note that they cross rated to the new rating and use that for the remainder of the bio.
In 2016 a few misguided senior officers (and a roundly criticized Master Chief of the Navy) tried to sell the Navy on using Petty Officer instead of their actual Rating, claiming that the Sailor's MOS was more important. I did 5 different jobs in 4 years, my MOS never changed. If I had been assigned based on my MOS, I would have been sitting side saddle on a low level position after 4 years even though I was a CTR2. Fortunately, the new Navy Brass realized they had rally screwed it up, and changed it back.
In doing a Bio for an old Navy guy, getting the Rating is as important as the Rank because it tells every other Sailor what he did, and in most cases how well he (or she) did it. In the Prefix space, I put the full Rating/Rank for people who actually retired from the Navy, because that is how the Navy would refer to him (or her) in correspondence.
You can see how the Official Navy does it on the CHINFO site: Here's a press release about an award upgrade: http://www.navy.mil/submit/display.asp?story_id=103612