Does sawing off a shotgun necessarily produce a "sawed-off shotgun"?
What's the difference between a shotgun that's been sawed off and a "sawed-off shotgun"?
The term "sawed-off shotgun" typically refers to a shotgun that's been illegally modified to make it shorter than a specified minimum length. In the U.S., federal law places a variety of strict controls on any shotgun with a barrel shorter than 18 inches, or one that's less than 26 inches long overall it must have a National Firearms Registration and Transfer Record, it requires a background check and a $200 fee prior to transfer, it may be transferred only though a federally licensed dealer, etc. (26 USC § 5845 provides some definitions.)
So if your shotgun has a 22-inch barrel and you use a saw to lop three inches off it, the weapon you now own isn't per se illegal in length and so, I suppose, wouldn't be what's generally thought of as a "sawed-off shotgun," though obviously that's literally what it is.
Why the prohibition against home shotgun surgery? Shortening the barrel of a shotgun makes it easier to conceal and in effect turns it into a particularly deadly handgun. In fact, in some jurisdictions, any device resulting from the sawing down of a shotgun legally qualifies as a handgun. A ne'er-do-well looking to pack some undetected heat may be prevented by local restrictions from simply going down to the nearest gun shop and buying a handgun, but with a shotgun, a hacksaw, and a vise he can create a potent firearm that's nearly as concealable. Accordingly, many jurisdictions prohibit outright the possession or transfer of illegally-modified shotguns, and the very act of sawing a shotgun barrel, to whatever length, may in some cases constitute illegal manufacture of a firearm.
Because firearm laws can vary dramatically from state to state, any sawing of shotguns is a potentially risky proposition. In short: if you've got a shotgun, learn to love it just the way it is.