Actually the .223 Rem has a higher case capacity than the 5.56 NATO round. The outside case dimensions are the same but the 5.56 NATO rounds walls are thicker which leads to a diminished overall case capacity (~.5 grains) and that smaller area in the case leads to the higher pressure inside the case walls and this increased pressure enables the 5.56 to discharge a higher grain projectile (NATO rounds are 62gr) at the velocity required by the military and have a tighter twist rate than the normal .223 in order to stabilize the heavier round. Also, the rifles designed to shoot 5.56 steel is designed to be safe with the increased pressure whereas the .223 is developed for the lower pressure rounds.
All .22x's are considered 22 calibers. The distinction is between rimfire and centerfire (.222, .222 mag, .223, 22-250, 220 swift, and a host of others). Most people understand that when someone refers to a 22 they are talking about the rimfire version being that most will refer to the centerfire version as the SAAMI cartridge chamberings.