Without context, it's difficult to comment.
For "extended casing," context is absolutely necessary, for instance. Without context, the term is meaningless.
"Casing," or "case," or "brass" all mean the same thing: The hollow rear portion of a cartridge, which holds the bullet, the propellant, and the primer.
The "bullet" is the projectile which exits the firearm and puts a hole into something.
The "propellant," also called the "powder" or the "gunpowder," is a fast-burning mixture or compound which generates gaseous matter when ignited.
The "primer" ignites the propellant when it is struck by the firearm's firing pin or "striker."
The "cartridge" is the self-contained assemblage of case, primer, propellant, and bullet. This is the "live round" to which you refer.
Each different size of cartridge is known by one or more different names. The name usually includes information about the bullet's diameter, and also some indication detailing which firearm, or firearms, the particular cartridge fits.
Examples: .38 Smith & Wesson; .38 Special; .38 Colt New Service (All of these are different, and not interchangeable.) The ".38" part indicates the bullet's approximate diameter, and the name part which follows the number explains the length and diameter of each different case.
A Colt's-made pistol can be found which fires the .38 Smith & Wesson cartridge, even though Colt's and Smith & Wesson are separate, competing gun manufacturers. Smith & Wesson designed the .38 Special cartridge, but Colt's (and almost every other pistol manufacturer) makes guns which require this particular cartridge.
The cartridge is not considered to be a part of the gun. It is merely something which is put into a gun, and not an intrinsic part of it. Therefore, it is not strange that a .38 Smith & Wesson cartridge would be put into a Colt's pistol that had been properly manufactured to accept it.
"Speer" is a manufacturer of ammunition. They produce bullets, as well as complete cartridges. A complete and unfired cartridge marked "Speer" could be of any size, to fit either pistol or rifle.
Usually, the cartridge case is marked on its base to indicate the "caliber" (that is, its size, and what chamber it fits). The only exceptions are the various lengths of the tiny .22 (or even .17) rimfire cartridge.
For "cartridge," click on: Cartridge (firearms) - Wikipedia, the free encyclopedia
For "rimfire cartridge," click on: Rimfire ammunition - Wikipedia, the free encyclopedia