Azumaya algebras

I have seen a few different definitions of an Azumaya algebra in the literature- for example, Wikipedia prefers the following one:

An Azumaya algebra over a commutative local ring $R$ is an $R$-algebra $A$ that is free and of finite rank $r$ as an $R$-module, such that the tensor product $A\otimes_R A^{op}$ (where $A^{op}$ is the opposite algebra) is isomorphic to the matrix algebra $\mathrm{End}_R(A) \sim M_r(R)$ via the map sending $a\otimes b$ to the endomorphism $x \mapsto axb$ of A.

On the other hand, for purposes of checking this definition, I have seen the following characterization used:

$A$ is a finitely generated projective $R$-module and for all maximal ideals $\mathfrak{m}\subset R$, $A/\mathfrak{m}A$ is a central simple $R/\mathfrak{m}$ algebra.

Is there an obvious way to show that these are equivalent?

EDIT: It has been noticed in the comments that the first definition specified "local ring"- this is too specific for the two definitions to be equivalent. I'm interested in what can be said when one removes local (and as Mariano points out, replaces free by projective) from the first definition and allows $R$ to be just a commutative ring.

An Azumaya algebra is by definition an álgebra $A$ which is separable over its center, that is, the multiplication map $m\colon A\otimes_{C(A)}A\to A,$ is a $(A,A)$-bimodule epimorphism that splits.