Great questions on a confusing issue.
Collectors tend to be picky about the quality of the coins they buy. Who can blame them since modern collector (semi-numi) coins usually have a high premium and even more importantly, their value is based in large part, on their condition (mintage numbers also can affect value as can other characteristics). So, let's talk about condition. What is generally considered top condition of a coin or medal? It's the quality of the strike (strong and full detailed impression) and the amount of imperfections on the coin / medal. Imperfections can include minor scratches, dents and dings, cracks, and even milk spots and other blemishes caused during or after production.
Coins and medals which have the fewest visible imperfections and the highest quality strikes are the one's that tend to be valued most highly by serious collectors who will pay big money for the finest examples...which can be pretty rare since imperfections can easily result from any number of things.
Generally, the coins that are struck earliest on a coin die will have the best impression because wear and tear on a die comes with every subsequent striking of the next blank (a blank is a term for the piece of metal the die strikes in order to create the impressions you see on the finished coin). that's why to some collectors, the earlier the coin was struck on a die, the more likely it will leave a better impression. But here's the rub. Logistically, when a mint produces a lot of the same coin, they generally stack them (in their boxes) high on shelves or even on the floor so that the coins struck first are on the bottom of those piles...in no particular order. When the coins actually leave the mint, the coins that are first struck may not be shipped within the first batch or even at all depending on the volume of orders and how many coins were minted. For very low mintage coins that come with numbered COA's, there is a way that a mint can actually match the lowest numbered COA's with the coins that were actually first struck.
Except for the "Numbered First Struck" designation (
http://www.ngccoin.com/coin-grading/scale-designations/other.aspx), all the other designations like "First Release", FirstStrike", and so on and so forth mean nothing at all....they are worthless and arguably very misleading. The "Numbered First Struck" designation is different because it has specific criteria which is supposed to be an verifiable accounting of the coins that are actually struck first by the mint meaning that those coins designated as "Numbered First Struck" are the one's which should have the best die impressions.
There's also the matter of how reputable the TPG (Third Party Grading) company is. Lots has been written about this all over the net and in forums like this but in general, in the US marketplace, NGC, PCGS, and ANACS are regarded as the most reliable.
So, the short answer is, in most cases, "First Release", "FirstStrike", and similar designations (other than the "Numbered First Struck" designation) are sales gimmicks and assure nothing at all in terms of what they insinuate to assure. What's more important in terms of the value of the coin / medal, is the grade it receives. I would argue that an MS 67 "FirstStrike" coin is less valuable to most collectors than an MS 70 graded coin that has no special 'First' designation. That said, there can be minor qualitative differences between similar coins receiving the same grade by the same TPG. That will likely have to do with the quality of the impression (strike) as opposed to the amount of imperfections. that's where the "Numbered First Struck" designation may be worth seeking out because coins labelled under this designation are supposed to have a paper trail from the mint assuring that the coins being graded with that designation are coins that were actually first struck with the dies by the mint.
This may not matter to most people now but to serious collectors it might and as graded coins appear to be gaining increasing support from coin buyers, it may become more important to more people in years to come that there's a grading designation which is alleged to have a paper trail to verify the claim of "first struck".
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