"There was no second twisting" Pete is correct
"The 1st picture shows what the Belgians would call Laminated. They were not "puddled"."
Pete is correct that Wire Twist/Plain Twist/Skelp was not puddled, Stub Twist was puddled c. 1850
The American Sportsman: Containing Hints to Sportsman, Notes on Shooting…Elisha Jarrett Lewis M.D. 1857 or Lewis’s American Sportsman
http://books.google.com/books?id=uXz6hcI5nWUC&dq=damascus+barrelsSTUB-TWIST BARRELS. “Before proceeding to manufacture them into iron, women are employed to sort and examine each stub, to see that no malleable cast-iron nails or other impurities are mixed with them. They are then taken and put into a drum, resembling a barrel-churn, through the centre of which passes a shaft that is attached to the steam- engine, which works the rolling-mill, bellows, &c. When the machine is put in motion, the stubs are rolled and tumbled over each other to such a degree that the friction completely cleanses them of all rust, and they come forth with the brightness of silver. The steel with which they are mixed, (generally coach- springs,) after being separated and softened, is clipped into small pieces, corresponding in size to the stubs, by a pair of large shears working by steam. These pieces are then, like the stubs, also put into a drum, in order to be divested of any rust they may retain, and are subsequently weighed out in the proportion of twenty-five pounds of stubs to fifteen of steel.”
“After being properly mixed together, they are put into an air-furnace and
heated to a state of fusion, in which state they are stirred up by a bar of the same mixture of iron and steel, until, by their adhesion, they form a ball of apparently melting metal. During this process, the bar has become sufficiently heated to attach itself to the burning mass, technically called a bloom of iron, and by its aid the whole is removed from the furnace to the forge-hammer, by which it is reduced to a bar of iron of far less weight than the original mass, the weight lost being wasted in the process of welding and hammering. From the forge it passes to the rolling-mill, where it is reduced to the size wanted. By this mode of manufacturing, the
iron and steel are so intimately united and blended that the peculiar properties of each are imparted to every portion of the mass, and the whole receives the degree of hardness and softness required. The process is admirable ; and the mixture is calculated to produce a metal the best fitted, under the circumstances, to answer the purpose of manufacturing gun-barrels of the best description.”
WIRE-TWIST IRON. This is the next quality of iron used in the manufacture of gun- barrels, and the mode of making the bar of wire-twist is thus described by Greener — “
Alternate bars of iron and steel are placed on each other in numbers of six each : they are then forged into one body or bar ; after which, if for the making of wire-twist barrels, they are rolled down into rods of three-eighths of an inch in breadth and varying in thickness according to the size of the barrel for which they are wanted ; if for Damascus, invariably three- eighths of an inch square. When about to be twisted into spirals for barrels, care must be taken that the edges of the steel and iron shall be outermost, so that, when the barrel is finished and browned, it shall have the appearance of being welded of pieces the size of wires the whole length of the barrel."
"The objection made to the wire-twist is that, owing to the iron and steel being
perfectly separate bodies, running through the whole thickness of the barrel, there is a difficulty in welding them perfectly, and of course there is a danger of its breaking across at any trifling imperfection. This objection is certainly well grounded, as many barrels break in the proving. I have myself seen a very strong barrel indeed broken across the knee without the slightest difficulty, while to all appearances it was perfectly sound. This is the reason why the manufacturers have ceased to make them, except for the American trade."
But things may have changed by 1900 (when Stubs were essentially unavailable)
Journal of The Federation of Insurance Institutes of Great Britain and Ireland, 1904 "Gun and Small-Arms Factories" by A.E. Patrick, p. 149-175
From 'Gun Barrels' starting on p. 159:
http://books.google.com/books?id=-QUtAAA...I6tLQ#PPA160,M1 The iron for the manufacture of sporting gun barrels was formerly made from finest scrap iron, such as old horse-shoes, nail stubs and the like. In preparing the metal for the old-fashioned laminated steel barrels, a number of scraps were collected of various proportions, the clippings of saws, steel pens, and scraps of best iron, which were placed for some time in a shaking barrel for cleansing, and then hand picked, in order that any pieces which had the appearance of cast iron might be removed. They were then cut into pieces of the same size, melted together, gathered into a bloom, and the mass placed under a tilt hammer, welded into a block of iron which was immediately rolled into bars. The bars were then cut into regular lengths, and the required quantity laid together and fastened into a faggot, this faggot was again heated in the furnace and hammered and rolled into rods of the size required by the barrel welders. The supply of fine old scrap does not now meet the demand, so at the present time the metal for gun barrels is made from a mixture of the best iron ores. The iron is made into rods, and subjected to hammering and rolling, which condenses the metal and increases the ductility and tenacity by elongating and densifying the fibres. The faggots are heated and welded seven times during the process of manufacture of the best barrel metal. The iron for the manufacture of gun barrels is made in square rods of various thicknesses for the best barrels, and in flat rods for plain twist or scelp barrels. To give the Damascus figure the square rods are first twisted, the operation being carefully overlooked to guard against one portion being twisted more rapidly than the other. This process is repeated until the rod is perfectly twisted and a regular figure in the barrel insured. It is this twisting of the rods that makes the difference between a best barrel and a common one.
All Damascus barrels must be made of twisted rods. Plain twist or scelp barrels are made from plain straight rods or ribands. 
