breach and breech are two different words with two different meanings. Calling the rear end of the chamber a "breach" is like calling the device used to lessen recoil at the front of the barrel a "muzzle break".
Breach is what Bud originally said - a break, rupture, or violation. As a verb it means to actively do one of those things. If used with regard to a gun, it's usually used incorrectly except when breaching a wall, door, etc. with a gun.
Breech is the back or lower part of something.
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Looks like "round" hasn't been covered.
It doesn't refer to only the bullet, it refers to a complete cartridge - case, primer, powder and bullet. I'm not sure we'll ever know the exact reason why a single cartridge is called a round. Maybe because one of the definitions of "round" is a complete something. For example, if you make one circuit of a prescribed course, you've made one round. In archery, if you fire a specified number of arrows at a target, you've fired a round.