As a gunsmith here in the UK.
The "Sleeved" marks are still used on new and old sleeving jobs this mark is still in use.
"lined" refers as others have said to teague liners. The idea was to save old damascus guns, by lining you could keep the appearance of finely wrought Damascus but have the bores restored to like new, by boring the original barrels that were shot out, out of proof, pitted or corroded or with terribly thin walls, essentially the guns were rebored and a liner was fitted into the tube through its full length, its always been my understanding that an epoxy of some kind held the liner in and filled any voids between liner and the barrel. In the short term it seemed to be an ideal solution, the guns passed proof, the damascus outer barrels were retained and the bores were like new.
Sadly the process was flawed, some of the issues were that once damaged or dented the dents were impossible to removed, also the nature of the two dissimilar metals sat next to eachother meant that they heated and cooled at different rates, expanding differently and causing the inner or outer barrel to "rivel" which was an issue beyond repair. The last thing was that the guns never quite handled as they should, the addition of thin wall lining tube through the length tended to give the guns a bit of a "dead weight" feel in the hands.
The marks "lined" are no longer used on shotguns because nobody now carries out this work in the UK. Though the mark may still be used at proof on rifle barrels that are lined but is never used to describe sleeving
The process is entirely different from sleeving, which is a topic heavily covered on these forums.
With sleeving the original barrels are cut off just forward of the lump, the chambers are bored out over size and new tubes are fitted and welded into position, the chambers are then re cut in the new tubes and the whole is struck off to match as closely as possible the original contours of the original barrels. There is good sleeving work and bad sleeving work but the end result is you end up with barrels that are like new inside and out... the handling characteristics and such depend on the quality of the job. When proofed these barrels are marked "sleeved" in the old days this used to be on the tubes themselves, now they are on the action flats with all the other marks.
