A quick and dirty...
Technically the earliest semiauto shotgun was a gas gun, the Clair sporting shotgun introduced in France in about 1895. The black powder of that era doomed it to unpopularity however.
Long recoil designs like the 'humpback' Browning Auto-5 and the Remington Model 11 are the 'old reliables,' and date back to 1903 when the first Browning came off the FN manufacturing line. The first Remington 11 was manufactured in 1905. Winchester had its own long recoil design, also known as the Model 1911, which was ... shall we say, constrained severely by Browning's patents. For instance, it had no bolt handle, and had to be initially operated by grasping a knurled section of the barrel and forcing it rearward. It was nicknamed 'The Widowmaker' in part as a result of this unfortunate method of operation.
Some people like the long recoil design better than anything else. In good condition and properly set up for the loads they're shooting, they are hard to beat for reliability, and they don't require a lot in the way of maintenance relatively speaking - just unworn reciprocating parts and springs and an owner who knows how to set them up and lubricate them properly.
Winchester's Model 50 was introduced in 1954 and was the first 'short recoil' operated semiauto. It featured the 'floating chamber' design of David Marshall "Carbine" Williams. The Model 50 was also a 'streamlined' receiver in the style that pump shotguns had shifted to several decades earlier, and it prompted others in the same style, including redesigned long recoil shotguns like the Remington Model 11-48.
Gas guns are much newer in the scheme of things, the first successful American gas operated shotgun was the High Standard Model 60, based on a Bob Hillberg design and introduced in 1954. Remington came along with the Model 878 in 1959 and the much better known Model 1100 in 1963.
The gas guns require a bit more cleaning and attention to keep running, but with modern designs takedown is simple and cleaning may be messy but is not that much of a chore.
The inertial designs came along later, around 1986 or so. The patent documents reference the earlier Williams floating chamber design, and the design is simple, relatively inexpensive to manufacture and easy to maintain in that it has fewer stressed parts. Like the long recoil guns, the inertial designs do require a solid 'base' to recoil against, or they may malfunction. Given a lighter bolt return spring, they might not feed troublesome ammunition as reliably as some other designs as well.
