I will assume that you know how to handle smooth differential forms on smooth real manifolds, this is a necessary prerequisite for understanding complex and holomorphic differential forms on complex manifolds.
(I won't mention commutative diagrams because I don't know how to draw these here, because I don't know which ones would be useful and because I'm confident that you'll come up with the ones you find useful yourself after you have understood the definition in the most simple setting.)
Suggestion: Let's look at the most simple situation, $\mathbb{C}$ as both a one dimensional complex manifold and as a two dimensional smooth real manifold $\mathbb{R}^2$.
On $\mathbb{R}^2$ we have a global chart, the cartesian coordinates, with coordinates denoted by $(x, y)$. In every point we have a basis of the tangential space $(\partial_x, \partial_y)$ and the dual basis of differential forms in the cotangential space $(d x, d y)$. Every (smooth) differential form can therefore be written (in this chart) as
$$
d w = f_1(x, y) d x + g_1(x, y) d y
$$
where we say that $d w$ is smooth if $f$ and $g$ are.
Now we can make a coordinate change in $\mathbb{C} = \mathbb{R}^2$ to the new coordinates
$$
z = x + i y
$$
and
$$
\bar{z} = x - i y
$$
This is an example of a biholomorphic mapping of an open set $V_2 = \mathbb{C}$ to $V_1 = \mathbb{C}$, which is therefore a change of coordinates on $\mathbb{C}$ (both as a real smooth and as a complex manifold). Now we can express every differential form $w$ again in the new coordinates:
$$
d w = f_2(z, \bar{z}) d z + g_2(z, \bar{z}) d \bar{z}
$$
The $f_2, g_2$ are related to $f_1, g_1$ (how?).
Motivation for the definition of "holomorphic differential form":
The differential $d f$ of a smooth function $f$ is a real smooth differential form. Similarily, one would like to define "holomorphic differential form" in a way such that the differential $d f$ of a holomorphic function $f$ is a holomorphic differential form. Since $f$ is holomorphic iff it depends on the coordinate $z$ only, one defines a holomorphic differential form to be a differential form that does depend on the coordinate $z$ only, that is $d w$ is holomorphic iff it can be written as
$$
d w = g(z) d z
$$
with a holomorphic function $g(z)$. This definition makes explicitly use of the global coordinate chart of $\mathbb{C}$ that we defined above. So, in order to show that "holomorphic differential form" is well defined on $\mathbb{C}$ seen as a complex manifold, we need to show that coordinate changes on $\mathbb{C}$ do not map holomorphic differential forms to non-holomorphic or vice versa.
A coordinate change in our example would be any biholomorphic map of an open subset of $\mathbb{C}$ to another open subset of $\mathbb{C}$.
Suggestion: Pick an easy example, write $d w$ in real coordinates and do a coordinate transform.
After that, have another look at the two definitions you cited.
Lemma. Every holomorphic function on a compact Riemann surface is constant.
Proof. Let $f:X \to Y$ be a nonconstant holomorphic mapping between (connected) Riemann surfaces, with $X$ compact. Then $f(X)$ is compact, therefore closed. But it is also open by the open mapping theorem. Therefore by connectedness $Y = f(X)$, and $Y$ is also compact. As $\mathbb{C}$ isn't compact, the claim follows. $\square$
Let's use the coordinate patches $(\mathbb{C},z)$ and $(\mathbb{C}^* \cup \{ \infty \}, 1/z )$. Since $f$ is meromorphic, it has only finitely many poles. We may assume that $\infty$ is not one of them (if it is the case, replace $f$ by $1/f$). Let $a_1,\ldots,a_n$ denote the poles of $f$. At the $i$th pole, $f$ has a principal part $$p_i(z) = \sum_{j=-k_i}^{-1} c_{ij}(z-a_i)^j$$
for some finite $\{k_i\}_{i=1}^n$.
Removing those yields the function $g = f - (p_1 + \cdots + p_n)$ which is holomorphic on all the Riemann sphere. But by the Lemma above, such a function must be constant. Therefore $f = g + (p_1 + \cdots + p_n)$ is rational.
Best Answer
You have two questions: how to change variables, and how to handle nodes. The case of changing variables, for hyperelliptic curves without nodes, is in chapter III.5.5 of Shafarevich's Basic algebraic geometry. The general case is on page 105 of Phillip Griffiths' Introduction to algebraic curves (China notes). The discussion of the nodes there needs supplementing with the argument from page 98, i.e. a proof that the derivative vanishes simply on both the separate branches lying over the node, rather than at the point in the plane, as suggested there in a footnote. Basically a fraction is holomorphic if the numerator vanishes as much as the denominator. Here the equation for the curve vanishes twice at the node so the derivative in the denominator vanishes once and is canceled by the vanishing of the adjoint polynomial. But you still have to finesse the point raised in Griffiths' footnote as mentioned above about the order of vanishing of the pullback to the normalization. Warm up on some specific examples.