Not sure from the Winchester perspective per se, but typically they were small bore, low-powered (primer only or 22 Shorts, 22 CBs, etc.) firearms designed to be shot inside the house. Some also call them ‘parlor’ guns.

While typically they were single shot, small framed guns, some European ones were quite ornate. Gallery pistols were even marketed.

Air rifles have all but replaced them .. less my .22 S-Shots I shoot in the basement range using Aquila Colibri (375 fps) and Super Colibri (~475-500 fps) ammo, both using primer compound only and a 20-grn bullet.

The same term "Gallery" rifle or gun is, unfortunately, used to describe all 3 types.

Schuetzen "gallery" rifles were normally found in .22 Short. Quite often these were high-end single shot rifles with all or many of the bells & whistles, sometimes engraved, sometimes not. They were used for practice & competition in the winter & in urban areas where use of a centerfire rifle was not practical. Large sums of $$ were wagered at the big-time urban matches. One that I am familiar with is an 1885 flat spring high wall with a #4 barrel, CCDST, a large nickle-plated buttplate and factory scope blocks.

"Shooting gallery" rifles were made for carnivals, circuses, amusement parks, etc. They were often slide action rifles as described, but I have used a friend’s Win. 74 in .22 Short with a shell deflector (presumably factory) and an eye-bolt in the stock so that the gallery proprietor could chain it to the counter. The 74 was remarkably accurate at 50 yards off the bench, even with open sights. Ammunition, mostly Shorts, came in "gallery packs" of 250 or 500 or more. Bullets were designed to disintegrate on impact & were advertised as "spatterless" or some such.

The last type was a light gun used for informal plinking or target shooting in a basement, cellar, etc.