1) About the second definition:  
$\alpha$) It is not true  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.  
  $\beta$) It is also not true that b) is equivalent to c):  
Take $R=\mathbb Z$ and $\mathbb Z\subsetneq M=\bigcup \frac {1}{p_1\cdots p_i}\mathbb Z\subsetneq \mathbb Q$ where $p_i$ is the $i$-th prime.  
 Then for all prime $\mathfrak p\subset \mathbb Z$ the $\mathbb Z_\mathfrak p$-module $M_\frak p$ is free of rank $1$, since $$M_{(0)}=\mathbb Z_{(0)}(=\mathbb Q) \quad  \operatorname {and}\quad M_{(p_i)}=\frac {1}{p_i}\mathbb Z_{(p_i)}$$ However the $\mathbb Z$-module $M$ is neither finitely generated nor projective (over $\mathbb Z$, projective=free)

2) 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.  

3) 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 $R$)  

4) 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).  
**Caveat:**  As shown in 1) above, the hypothesis that $M$ is finitely generated is vital.