overuse of the word "Damascus" is one of my pet peeves...
so i hope I don't bore you all (oh dear, barrel making puns)
First there was Damascus the city, and the "Damascus steel" that was traded there. "True" Damascus steel is a special high carbon cruicible steel with trace quantities of vanadium which cause a special crystal pattern to appear if it is worked just so. It actually came from India and "true" damascus is usually called "wootz" steel these days.
The "true" damascus was so highly prized in medieval europe that people started imitating the patterns by "pattern welding" combinations of steel and iron wire and strip together. This technique was developed by the vikings and celts for swords and eventually used in gunbarrels. You really should call them "false" damascus barrels, or "pattern welded" barrels.
To add to the confusion there were barrels which were even "falser" with painted or etched patterns in plain iron or steel barrels.
And of course silver and gold inlay was referred to a s "damascene work" (and the patterns "arabesques") and patterned fabric as "damask" (because the originals were from Damascus the city, not because the patterns of inlay or silk looked like wootz steel)
clear as mud eh?
and as for quality, well you find good and bad in 'pattern welded" and fluid steel barrels. I worked on a set of cast steel barrels recently and was a bit allarmed to find some pinhole type pitting in the breech.
the gun is now a wall hanger
if in doubt, get the barrels tested. X ray, magnafux, or dye penetration test