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.

1. $K_X$ is nef.
2. 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 1 and 2 are equivalent definitions? In particular, why 2 implies 1. Does exceptional divisor contain extremal ray?