How many models depends on what you mean by model. In actual production models, essentially there was only one. The variations were limited and were not really considered different models.
By "grooved" I assume you mean fluted. While those are in the under 8000 range, there were probably only about 4000 actually made. (I don't know how many were made after the Italian repros and fake marking stamps turned up.) The fluting of the cylinders was probably done for the usual reason, reduction in weight, but was dropped when some cylinders failed. The factory records don't seem too reliable on this; Wilson says the fluted guns were recorded as "cavalry models" but later researchers found fluted guns not so recorded, and unfluted guns marked as cavalry. One problem is that when a cylinder failed, the factory replaced it with a new, unfluted cylinder, serial numbered appropriately. (Will the real fluted Armies, please stand up?)
The guns purchased with stocks would have had the frame studs for the stock, but the stock will fit and work quite well without them. (In Wilson's Book of Colt Firearms, there is a picture (page 165) of a cased presentation 1860 with stock, but without the stock studs.) The norm iis for the recoil shield to be cut for stock but without the stock studs, and the later guns do not even have the flat hammer screw, indicating there was no intention of using a stock.
Guns without the frame cuts are rare and I have never personally seen one, though I have seen an 1851 Navy with stock cuts in the frame, also a rarity. AFAIK, all the steel backstraps were cut for the stock. Brass backstraps are found only in the low number range; AFAIK, steel backstraps were never silver plated from the factory.
Since those guns are so common even today, some people like to use them to turn out "rare" guns that Colt never made. That did not begin last week; it has been going on for decades and pre-1960 fakes are all around, having by this time acquired the look of age and being accepted as "rare variations" even by experts.
Jim
