I'm starting to learn the minimal model program. It seems there are two definitions for a variety $X$ with only terminal singularities to be minimal:

- $K_X$ is nef.
- Every birational morphism from $X$ to $Y$ must be an isomorphism, where $Y$ is another variety with only terminal singularities.

Suppose $X$ is a smooth minimal model of dimension greater than 2. Why are 1 and 2 equivalent definitions? In particular, why does 2 imply 1? Does the exceptional divisor contain an extremal ray?