cv.complex variables - what is the formal definition of multi-valued holomorphic function?

It seems this notion causes some confusion not only on me.
Let me give my own definition, by taking into account with yours.

Let $X$ be an analytic manifold, and let $pi:tilde{X}mapsto X$ be some universal covering of $X$.　A multivalued holomorphic function on $X$ is a holomorphic function on $U$, where $U$ is some contractible open subset of $X$, if the function $picirc f$ on some component $mathcal{C}$ of $pi^{-1}(U)$ can be analytically extended to $tilde{X}$. Denote this function by $tilde{f} _mathcal{C}$

Two multivalued holomorphic functions $(f,U)$ and $(g,V)$ are equivalent, if there exists some component $mathcal{C}$ of $pi^{-1}(U)$ and $mathcal{D}$ of $pi^{-1}(V)$, such that $tilde{f}_mathcal{C} = tilde{g}_mathcal{D}$.

This definition doesn't depend on the choice of universal covering.

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Edit:

Here is a more rigorous definition：

Definition. A multivalued holomorphic function on $X$ based at $x$ is a function $f$ in $mathcal{O}_x$, such that one component of $pi^* f$ can be analytically extended to $tilde{X}$, where $pi: tilde{X}mapsto X$ is some universal covering.

Denote the ring of multivalued holomorphic functions on $X$ based at $x$ as $tilde{mathcal{O}} _x$, which is a subring of $mathcal{O}_x$.
Any loop based at $x$ gives an action on $tilde{mathcal{O}} _x$, which is essentially the monodromy.

Claim. $tilde{mathcal{O}} _x$ is locally free $mathcal{O}(X)$-module with rank the order of fundemental group.

It is essentially equivalent to the defintion that a mutilvalued fundtion is defined as function on covering space. But i feel it is more intuitive and doesen't depend on the choice of covering space.

Under this definition, I can make sense of the solution of mulivalued function of some analytic differential equation, which is actually my original motivation to ask this question.