First, Damascus and laminated barrels were made because prior to around 1890 factories were not able to drill a straight hole in a piece of solid iron or steel as long as a shotgun barrel. They continued past 1900 mainly because they were the product of cheap labor and many people liked the looks of a Damascus barrel. (Even after barrels were made from solid steel, some makers turned out fake Damascus to satsfy that market.)
The best way to determine if the barrels are Damascus is to look at the markings (many are marked "Damascus" or "Laminated") or to look at the barrels for signs of a spiral pattern. Damascus barrels were made by twisting thin bars of iron and steel together, then welding them by heating white hot and hammering them. Then the resulting bars were thinned to the proper size, again by heating and pounding, then wrapped around an iron bar called a mandrel.
Once again, the bars were heated and welded together by pounding on them. Once the barrel was reamed out, finished and polished, it looked almost like solid steel, but actually it was the product of thousands of welds. Laminated barrels were made much the same way except that the welder started with small iron and steel plates rather than with bars, bending them around the mandrel.
Some Damascus barrels were very well made. English made barrels were always good, and so were the better class of Belgian barrels. There were some made in the U.S. and those were good, also. Unfortunately, most of the Belgian guns and many U.S. guns were not top quality. They were utilitarian guns and filled the demand for shotguns in a day of plentiful game and no limits.
But time and corrosion have taken their toll on many of those guns. Powder fouling and primer salts have worked their way into the microscopic and multitudinous crevices and joints in the barrels, eating them out from the inside, even if the outside and the bore look shiny and clean. Some people insist that Damascus barrels are OK with black powder, but they do let go with even black powder and smokeless powder is an absolute no-no.
The main problem with those barrels is that they almost always let go out where the barrel becomes thinner, which is exactly the place the shooter places his off hand. The result is that a blown barrel usually takes some fingers along with it.
FWIW, H. J. Sterling was a Crescent brand name. It may have been just a nice sounding name, or it may have been the name of a hardware store, a chain, or distributor, as it was common at that time for even single stores to order guns with their name on them, and if the order was fairly large, Crescent would oblige. IIRC, at one time, an order for as few as 20 guns would be marked with whatever name the customer wanted.
Jim