There is no difference. If $M$ is locally free of finite rank, then $M$ is of finite presentation (and projective).

Take a partition of unity $f_1,...,f_n$, such that $M_{f_i}$ is free over $R_{f_i}$. Since $R \to R_{f_1} \oplus ... \oplus R_{f_n}$ is faithfully flat, it suffices to show the properties for $M_{f_1} \oplus ... \oplus M_{f_n}$, which is very easy.

There is no difference. If $M$ is locally free of finite rank, then $M$ is of finite presentation (and projective).

Take a partition of unity $f_1,...,f_n$, such that $M_{f_i}$ is free over $R_{f_i}$. Since $R \to R_{f_1} \oplus ... \oplus R_{f_n}$ is faithfully flat, it suffices to show the properties for $M_{f_1} \oplus ... \oplus M_{f_n}$, which is very easy.

Definition (2) is prefered because it reveals the geometric content: classification of line bundles.