Meromorphic differentials and the pullback. In Diamond & Shurman's A First Course in Modular Forms Section 3.3, the authors naively treat so-called "meromorphic differentials". It seems that the spaces they denote $\Omega^{\otimes n}(X)$ is really the $n^{\text{th}}$ symmetric power of so-called meromorphic differentials, denoted $\Omega^1(X)$.
Here is my first question. What is $\Omega^1(X)$? This question has the answer "It's a section of the n-th symmetric power of the bundle of meromorphic differentials" but for me this simply raises the question. What are meromorphic differentials?
I believe I know what $\Omega^1_{\text{hol}}(X)$ (holomorphic differentials) is, it should be defined just as smooth differentials are for real manifolds, that is $\Omega^1_{\text{hol}}(X)$ is simply sections of the cotangent bundle.
Now onto the pullback. Locally, the authors define the pullback as follows: if $\varphi:V_1\to V_2$ is a holomorphic mapping of open subsets of $\mathbb C$. Then
$$\varphi^*(f(q_2)(dq_2)^n)=f(\varphi(q_1))(\varphi'(q_1))^n(dq_1)^n.$$
I'd like to understand this in the context of differential geometry. My understanding (admittedly almost exclusive to real manifolds) is that if $\varphi:X\to Y$ is a holomorphic mapping of Riemann surfaces, the pullback $\varphi$ on $\operatorname{Sym}^n\Omega^1(Y)$ at a point $P\in X$ is
$$\varphi^*_P(\omega_1\cdots\omega_n)(X_1,\dots,X_n):=\omega_1(\varphi_{*,P}X_1)\cdots\omega_n(\varphi_{*,P}X_n)$$
where juxtaposition $\omega\eta$ denotes the symmetric product, $\varphi_{*,P}$ denotes the push-forward, and $\frac{1}{2}(\omega\otimes\eta+\eta\otimes\omega)$ and $X_1,\dots,X_n\in T_PX$.
Here is my second question, somewhat contingent on the first.
How do I recover the local expression from Diamond and Shurman from the differential-geometric one? Am I even expressing the naive definition correctly in the language of manifolds in the first place?
So far, I have only studied vector bundles in the context of differentiable manifolds. I hope to (very soon) start studying quasi-coherent sheaves, but I don't understand this framework yet, so please answer in the context of real and complex manifolds and bundles, though some indication for how this works for invertible sheaves would also be appreciated as I should understand this a bit later.
 A: Let me elaborate on the comment above and turn it into an answer.
There are many ways to define meromorphic differentials, maybe the simplest of which is "rational sections of the cotangent bundle". To be more precise, if $V \subseteq X$ is an open subset, and $U \subseteq V$ is such that $V \setminus U$ consists of isolated points, then $\omega \in \Omega^1_{hol}(U)$ is a meromorphic differential on $V$ if it has poles along $V \setminus U$.
Alternatively, one could simply define a meromorphic differential form on $V$ using charts - defining meromorphic differential forms on $\mathbb{C}$ as $ f(q) dq$ with $f$ meromorphic, and say that $\omega$ is a differential form on $V$ if for any chart $\phi : W \rightarrow V \subseteq \mathbb{C}$, $\phi^{*} \omega$ is meromorphic. This is the approach taken in Diamond and Shurman.
The space denoted $\Omega^{\otimes n}(X)$ is not the $n$-th symmetric power, but the $n$-th tensor power. When we have two line bundles, we can tensor them to obtain a new line bundle. We do that $n$ times for the line bundle of meromorphic differentials. In more explicit notation
$$
(dq)^n = (dq)^{\otimes n} = (dq)\otimes \ldots \otimes (dq)
$$
If $\varphi : X \rightarrow Y$ is holomorphic, then pulling back satisfies
$$
\varphi^*(\omega_1 \otimes \ldots \otimes \omega_n) = 
\varphi^*(\omega_1) \otimes \ldots \otimes \varphi^*(\omega_n)
$$
and we obtain
\begin{align*}
\varphi^*(f(q_2)(dq_2)^{\otimes n}) &= \varphi^*(f(q_2)dq_2) \otimes (\varphi^*(dq_2))^{\otimes n-1} \\
&= (f(\varphi(q_1)) \varphi'(q_1) dq_1) \otimes (\varphi'(q_1)dq_1)^{\otimes n-1} \\
&= f(\varphi(q_1))(\varphi'(q_1))^n (dq_1)^{\otimes n}
\end{align*}
where here we have used the chain rule to pull back each differential.
