Dear Kwan, 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 nilpotent radical of $A$ is a prime ideal) 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 ( P\otimes_A K) $ . This is the definition given by Matsumura in his book Commutative Rings, page 84. It corresponds to the maximum number of elements of $M$ which are linearly independent over $A$.
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,... Geometrically it corresponds to the minimum numbers of global sections of $\tilde{M}$ which generate this sheaf at each point of $spec(A)$. Elementary fairly well known example: every ideal of a Dedekind domain can be generated by two elements.