Jesse, I agree that's a question, but it's not the only one. The ads tell us that some pipes were marketed as Barling straight grains as early as 1912 (and I have many subsequent ads demonstrating the same thing over the ensuing decades), and the article demonstrates that there was an established market for straight grained pipes by 1904. But as you say none of that speaks to nomenclature.
For what it's worth I suspect Barling's straight grains were stamped as such; after all, YOW was likewise a classification, dated from around the same time, and appeared in ads and on the actual pipe. Even a century ago people liked to know they were getting what they paid for; I imagine guys buying a straight grain wanted it labeled as such.
Moreover we also know that Barling was making a big deal of the value of a beautifully straight grained pipe. I just found this cool ad (clearly a marketing stunt, except of course Barling would never stoop to marketing) from the Perthshire Advertiser, dated Nov 26 1927, placed by our old friend Charles Rattray:
I think that demonstrates enough economic value, and associated cachet, that it suggests straight grained pipes would be marked as such.
But at the end of the day all this, of course, is pure speculation. I don't think we'll ever know for sure unless we uncover another early catalog, or a hallmarked straight grain pipe from that era.
By the way, I found the Barling straight grained pipe mentioned in the Rattray's ad. I knew the family had already refused £100 pounds, so I offered Monty's heirs £101 and they took it.