Agree that those look more like snipe than woodcock. My guess as to why you don't see more woodcock engravings is that woodcock, at least historically, have been a bonus bird and not the sole game species hunted, let alone dedicating a specific gun purchase to.
Maybe our UK friends could chime in on this, but as I understand it, woodcock are almost always a rare bonus in a days hunt on their side of the pond. In the US, historically, woodcock were also a welcome bonus during a days hunt for grouse. This is just my opinion, but the specific targeting of woodcock in the US, that is dedicating specific guns, and other various gear just to woodcock hunting, is a fairly recent phenomenon (20th century?).
Alternatively, the hunting of snipe is a very specific day of hunting, where snipe would be the main target species and other game would be the bonus. For a dedicated snipe hunter, it would be very reasonable to order a gun just for that purpose, which might include ordering one with game scene engraving of your main quarry.
Just thinking out loud, but feel I am not too far off.
