In the theory of Riemann surfaces, for a meromorphic function $f$ on a Riemann surface $X$, we define the divisor of $f$ to be the function $D$ which maps $$f \longmapsto \sum_{p \in X} \text{ord}_p \cdot p,$$ where $\text{ord}_p$ denotes the order of the root or pole of $f$ at $p$.

In the context of Stein manifolds however, we define a divisor to be an element $d \in H^0(X, \mathscr{D})$, where $\mathscr{D}$ is the quotient sheaf of $\mathscr{M}^{\times}$ by $\mathscr{O}_X^{\times}$, where $\mathscr{M}^{\times}$ is the sheaf of germs of invertible meromorphic functions and $\mathscr{O}_X^{\times}$ is the sheaf of germs of invertible holomorphic functions.

Are these definitions equivalent? If they are, I can't see why.

Note that this does seem like a question that should be posted on Math.StackExchange, but I have found little to no luck asking questions of anything north of the elementary theory of analytic functions of several complex variables on that site.