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It is not true in the second definition that for an arbitrary ring a) is equivalent to c).
Indeed Bourbaki in Algèbre commutative, Chapitre II, Exercices §5, 12) c) exhibits a ring $B$ and a projective module of rank $1$ over $B$ which is not isomorphic to an invertible fractional ideal of $B$. This does not contradict Eisenbud's Theorem 11.6 because $R$ is explicitly supposed noetherian there.

I think the only reasonable definition of $\operatorname {Pic(R)}$ valid for any commutative ring is to define it as the Picard group of the affine scheme $X=\operatorname {Spec}(R)$.
As is the case for any locally ringed space $(X,\mathcal O_X)$ the Picard group consists of isomorphism classes of locally free $\mathcal O_X$-Modules of rank one.
This is exactly the definition used with much success for general, non-affine, schemes but also for topological spaces, differential manifolds, etc.

In our special case $X=\operatorname {Spec}(R)$ the definition in 2) translates in purely algebraic terms to:
$\operatorname {Pic(R)}$ consists of isomorphism classes of $R$-modules $M$ such that there exist finitely many elements $r_1,\cdots,r_n\in R$ with:
$\alpha$) $\sum Rr_i=R$
$\beta$) $M_{f_i}$ is a free $R_{f_i}$-modules of rank $1$ for all $i$.
(Remark: $\beta$) implies that $M$ is finitely generated over $R$)

The modules $M$ in 3) can also be characterized as the finitely generated modules over $R$ such that equivalently:
i) $M$ is projective of rank $1$
ii) The modules $M_\frak m$ are free of rank $1$ over $R_\frak m$ for all maximal ideals $\mathfrak m\subset R$
iii) The canonical $R$-linear map $M\otimes_RM^*\to R:m\otimes \phi\mapsto\phi(m)$ is bijective
Note that these are pleasant algebraic characterizations of the modules generating $\operatorname {Pic(R)}$, but the better, conceptual definition is that given in 2) and 3).
A final caveat:
The implication ii)$\implies$ i) above is false if one does not assume that $M$ is finitely generated over $R$.
A counterexample is to be found in Bourbaki Algèbre commutative, Chapitre II, Exercices §5, 7) b)