What's wrong with defining the rank of a finitely generated module over any (commutative) ring to be just the smallest number of generators? All books I know define rank only locally this way. But why not define it globally?

since your profile says you are interested in Algebraic Geometry, here are geometric considerations that might appeal to you. Consider a projective module $P$ of finite type over a commutative ring $A$. It corresponds to a locally free sheaf $\mathcal F $ over $X=Spec(A)$. The rank of $\mathcal F $ at the prime ideal $\mathfrak p$ is that of the free $A_{\mathfrak p}$module $\mathcal F_{\mathfrak p}$. The rank is then a locally constant function on $X$ and if $X$ is connected (this means that the only idempotents in $A$ are $0$ and $1$) it may be seen as an integer. If $A$ is a domain, then $X$ is certainly connected and has a generic point $\eta$ whose local ring is the field of fractions $\mathcal O_\eta=K=Frac(A)$. The rank of $\mathcal F $ or of $P$ is then simply the dimension of the $K$ vector space $P\otimes_A K$. Actually, if $A$ is a domain, this formula can be used to define the rank of any $A$module $M$ (projective or not, finitely generated or not) : $rank(M)=dim_K ( M\otimes_A K) $ . The minimum number of generators of $M$ (which started this discussion) is quite a different, but interesting invariant, which has been studied by Forster, Swan, Eisenbud, Evans,... Bibliography 

