So called specimens from the early days of the mint (1793-1795) are highly speculative. Generally, these pieces supposedly have better strikes, superior planchets, and may have reflective surfaces. The mint did not make proof coins at this time, but some think that they may have made special pieces for presentation to dignitaries, or as a proof of concept for a design. There isn't much documentation or evidence to justify calling them specimen pieces, as far as I know.
Good question. Better question ! I guess it depends on who ya listen to. Over the years the TPGs have kinda gone one way at one time and then changed and gone another way - calling the exact same coins by different names/designations. PCGS has this on their web site - Specimen Term used to indicate special coins struck at the Mint from 1792-1816 that display many characteristics of the later Proof coinage. Prior to 1817, the minting equipment and technology was limited, so these coins do not have the “watery” surfaces of later Proofs nor the evenness of strike of the close collar Proofs. PCGS designates these coins SP. Not that many years ago a Specimen coin was a special coin struck for special reasons, they were not intended as Proofs, not intended for circulation, not intended to be used for anything really other than as an example for the mint to examine. In other words it didn't have anything to do with the finish on them. That said, yes the mint did make presentation pieces, and they did that every now and then for a lot of years. But presentation pieces are different than specimens because presentation pieces were minted with the intention of giving them to somebody special. They even made special boxes for them. The practice of making presentation pieces dates back to long before the US Mint even existed, and back then they were called presentation pieces.
The 1965-67 SMS sets would be considered specimens. Specially prepared dies, but planchets are not polished and coins are not double struck.
2005-2010 satin finish coins are not graded as specimen. They are given a business strike grade designation. I believe that is misleading. You see people trying to sell these satin finish coins as if they are normal business struck coins intended for commerce. The satin finish coins were only supposedly available in mint sets. There were circulating coins that sure looked satin finish. The whole satin finish grading always seemed a little fishy. Who was trying to boost collecting/buying coins more? The TPGs or the mint? If the grading companies couldn't tell the difference between finishes, then, they either shouldn't grade coins or all satin finish looking coins should have received an SP designation.