Only one or two of my Barkies say "1st prod run", a couple just say "prototype". I think that, once the first prod run is out, and the knives go over well, or well enough, they just start making them without any more inscription. Maybe it's related to knife collection, similar to an inscription like "1 of 300" as an example.
There really wouldn;t be any collectors value to knives marked "second prod run", at least not that I can see.
Even if my take on it is right, I don;t treat any of my BRKT knives as collectors, regardless how they're inscribed.