I think the small "Detective" over the small "Special" stamping cannot be seen well in the above picture, but it is there. Enlarge one of the other pictures of the barrel in the listing which shows it more clearly. The OPs picture does not appear to have "Detective Special" in 2 lines stamped on the barrel, just "38 SPECIAL" of the same size. The Serven book's picture of a first issue DS also clearly shows "Detective Special" in two lines stamped on the barrel, with the taller "38" to the left of them.

It also appears from the OP's picture that the barrel length is a little shorter than that of the real first issue. I have no doubts that at least the OP's barrel in not factory original.