Bending wire versus cable
the minimum bend radius can serve two purposes.

the first is to protect the inner "conductor". on fiber optic cable and very thin copper cables, going below the minimum bend radius can cause the "conductor" to break.

the second is to protect the sheathing. on heavier gauge wire such as electrical wiring, going below the minimum bend radius can cause the sheathing to break, exposing the inner conductors. when you get to the end of the wire (as in a switch box) a tight bend causes the sheath to retract from the exposed end instead of just breaking.

breaking the inner conductor is not as much of a concern with electrical wiring because, at these diameters, it generally takes repeated flexing to cause a failure in the conductor itself, and these types of cables are meant to be installed in fixed environments.