AFAIK noone here called it a "pigeon gun". Pigeon guns were specially designed for the sport of life pigeon shooting, as described by Greener. They usually have long, tightly choked barrels and more often than not no safeties. In a way they were similar to modern Olympic trap guns.
Churchill designed his XXV as Game Guns. A "British Game Gun" is built for "Formal Shooting" in the first place, that is shooting at driven pheasants, partridge or Scottish grouse. This ideally is shooting at birds flying over a line of "guns", so most shooting is upwards. Game guns are also used for "Rough Shooting" in Britain, that is fur and feather hunting as you probably know better: rough shooting is following your dog, be it a spaniel, pointer or setter, through fields and brushes, shooting at birds and/or hares and rabbits your dog flushes.
Sitting in a blind near water and shooting at ducks and geese is "Waterfowling" in British parlance, requiring a "Fowling Gun", most often of less quality and embellishment than a "Game Gun", but shooting heavier loads for more reach.
Carrying a rifle and looking for deer, either walking through the forests or sitting on a high seat, is "Stalking".
"Big Game Hunting" is travelling to the former colonies in Africa or India and shooting elephant, buffalo, Lion or tiger.
None of these sports is simply "Hunting". When an Englishman goes "Hunting" he puts on a red jacket, mounts a horse and chases after hounds and foxes.
These subtleties were explained to me many years ago by a friendly clerk at the largest London bookshop when a dumb Kraut asked for books on hunting.