Musket...
A heavy military arm dating back to the matchlock era. With the addition of a bayonet, they are a pike that shoots.
Fusil....
This French origin term is sometimes found as, fuzee, fuzil and fuke. A fusil is a light Musket. The term originates with the adaption of the flintlock ignition system in the second half of the 17th Century. At that time Matchlock Muskets were relatively heavy and less mobile. The new flintlock fusils were more efficient, had a higher rate of fire and lighter. These early military flintlocks were excellent for specialized light troops, Fusileers. As the turn of the 18th Century approached, most armies had adapted the flintlock fusil as the main battle weapon. The Dutch pattern light musket or fusil eventually becomes the Brown Bess. Thereafter "fusil" refers to longarms lighter than standard military muskets. Officers Fusils could simply be scaled down versions of the standard musket. The term is also used for non military guns such as trade "fusils" and hunting...fuzees and fukes.
17th Century a Fusil is a light military flintlock...
18th Century a fusil is a light musket or smooth bore hunting arm.
17th and early 18th Century trade fusils, buccaneer, Dutch, French and English trade fusils are commonly found with a rear sight. Another aspect of the fusil is that although lighter than a military musket, it tends to be heavier than a fowling piece.
Fowling Piece or Gun...
A smooth bore arm designed primarily for wing shooting. These tend to be light, graceful and have long barrels. The line fusil and fowling gun can be quite blurred, especially in North America where the fowling piece would be commonly loaded with ball thus ...more like a general purpose fusil. A fowler is not a gun but one who hunts fowl.
Shotgun...
This general term primarily refers to civilian guns such as fusils or fowling guns. By the 19th Century shotgun was a general term for civilian smoothbores replacing fusil and fowling piece. It's possible the term dates to the 18th Century. Bakeless says in his 1935 biography of Boone, " The Shawnee took the hunting parties' hides, horses and rifles. They then provided them with an old shotgun(likely a trade Fusil) and hunting pouch with a warning, Never to return. This is likely from the Draper Papers.
Smooth Rifle...
This is a firearm that has all the characteristics of a rifle but with a smooth bore. These seem to be most prominent in PA. Lehigh Valley Rifles are commonly found smooth bored. On occasion some Lebanon and Lancasters are found smooth bored.
Buck and Ball gun...
A century earlier these would have been called fusils. These are smoothbored guns that can have a mix of rifle and smoothbore architecture, for instance round barrels, rear sights,crescent butts and cheek pieces. Primarily....early 19th Century.
Smoothbore...
This is a term for all the above.
Fort Stoddart (the Choctaw Factor 1805)...." Smoothbores are virtually unsaleable to the Choctaw, They want Rifles."