Open source indicates 2006 is when SWCC got the SB designations. Prior to '06 SWCC got NSW ratings and used regular Navy titles. SWCC didn't become formally "closed loop" from the regular Navy until '96 (but may have had an informal culture of retaining members rather than rotating out), and a need for what would become the SB designations was recognized, but it took a decade including a half decade of a much higher operational tempo for the Navy to finally set out the SB.
Of course the 90s in T2K weren't our 90s, so who knows what definitively would have happened since the country was more on a path to "total war" in 2TK than the more SOCOM intensive warfare that's marked the U.S. military of the 21st century.
Mechanically, the YZE T2K lifepath I think does a good job creating "regular military" character. The Special Operations life path options are a little thin, but I think understandably balanced to not tip the scales too against other characters who don't get SOCOM exposure (modern gaming lifepaths tend to frown on wildcard boons to stats and skills). The RDF Sourcebook for the original T2K was where you'd find the options for Special Operations backgrounds, and they were much more formidably skilled than your regular T2K soldier. Part of that's because the IRL pipelines for operators take literal years longer than regular military (except maybe SWCC prior to 96?). This isn't represented well in the present T2K lifepath, but the skill system isn't granular enough for it to ultimately matter. Plus it allows you to play a SEAL, a PJ, or a Green Beret pretty much on say so if you land the block in character generation.
In saying all that, I think it's pretty much up to your individual table what ranks or titles a character might have had before coming on a SEAL team. As for your question about whether the character would still be SB or NSW equivalent after making it through SEAL training. No, that's no different than no longer being an MP after becoming a Ranger, if that makes sense. You're your present job. What you used to be might've helped get you there, but it's not you.
Just barely starting to hit the forums, so I'm playing catch- up =)
Basically, prior to 2006, a Sailor would be whatever his rating was, i.e. a Corpsman would be an HM3 (SEAL), and a Hull Technician would be HT3 (SEAL). Also, in the T2K timeline, OP can fudge the Combat Arms requirement as the Navy did not have what is considered Combat Arms outside of the Special Warfare Operators, the closest being a Gunner's Mate (GM). The majority of Operators came out of (in T2K terms) Combat Support and Combat Service Support - and no, there weren't any cooks who were Seals... lol