The most common usage for carbine is usually a shorter rifle. The earliest "modern" usage I've read of were the lever action Winchesters which were made with long "rifle" barrels and shorter "carbine" versions.
This usage of "carbine" continued through the 20th century.
The Jeff Cooper definition of rifle and carbine differs. The Colonel's "rifle" definition is a long arm shooting a full power cartridge. The definition of "full power cartridge" is fuzzy but usually something in use by the world's military's in the early half of the 20th century as a minimum. Examples are .30-06, .308, .303 British, 8mm Mauser, 6.5x55 Swede.
Another way to look at full power rifle definition by Cooper's side is something that can strike a telling blow to whatever you can see. I know that sounds vague. Read a lot of Cooper's stuff and it makes more sense.
The Colonel's definition of "carbine" is something shooting a caliber powered down, or inferior to the full power stuff. Note you could argue that .308 is a carbine round as it was sort of a cut down .30-06, but nobody in the Jeff Cooper school of thought seems to look at it that way as .308 is still considered pretty full power.
OK. 8mm Kurz or whatever the ole Wehrmacht Sturmgewehr shot was an example of a "carbine" caliber. Other examples are .223, 7.62x39 (AK-47 ammo), 5.45xsomething mm (AK-74 ammo), .30 M1 Carbine, etc.
The Jeff Cooper school of thought is pretty hard on carbines such as the AR-15/M-16, AK, M1 carbine, etc. I myself think that Jeff Cooper "rifles" are best for extended range shooting (beyond 25-50 yards). Cooper definition "carbines" are best for the close range multiple target engagement, such as in final infantry assault on a position.
I don't think the 2 types replace each other, they complement each other. Trouble is for the individual, which one do you carry?? I personally think Cooper carbines are better suited to defensive situations and the Cooper rifles are better for offensive "I'm looking for trouble" engagements.
Sorry for the long post. I got really into this one.
Edmund