I'm following Gathmann's notes (from 2021) and I got to the section on regular functions (on affine varieties). He gives the following definition, which I believe is pretty standard:
Definition 1. Let $X$ be an affine variety (not necessarily irreducible), and let $U$ be an open subset of $X$. A regular function on $U$ is a map $\varphi\colon U \to K$ with the following property: for every $a\in U$ there are polynomial functions $f,g\in k[X]$ with $f(x)\neq 0$ and $\varphi(x) = \frac{g(x)}{f(x)}$ for all $x$ in an open subset $U_a$ with $a\in U_a\subseteq U$. The set of all regular function on $U$ will be denoted $\mathcal{O}_X(U)$.
However, I don't find this definition very intuitive, specially when compared to other texts. For instance, an older version of the notes (from 2002), or Mukai's book, give an alternative one, which I think is more natural:
Definition 2. Let $X$ be an irreducible affine variety. As its coordinate ring is an integral domain, it has a well defined field of fractions, called the set of rational functions on $X$, which we denote as $k(X) := \mathrm{Quot}(k[X])$. Given a point $P\in X$, the set of rational functions on $X$ defined at $P$ is $$\mathcal{O}_{X,P} := \left\{\frac{f}{g}\in k(X)\colon g(P) \neq 0 \right\}.$$ Given an open set $U\subseteq X$, the set of rational functions of $X$ defined on $U$ is the set $$\mathcal{O}_X(U) := \bigcap_{P\in U} \mathcal{O}_{X,P}$$.
I'm aware these sets are usually called the local ring of $X$ at $P$, and the ring of regular functions of $X$ on $U$, respectively. My first question is the following:
Q1. Do these two definitions actually coincide?
This question is prompted because the texts tend to emphasize the fact that the local representations of a regular function may vary from point to point, ie. there is not a global representation of a regular function on all of $U$, necessarily. The thing is I'm failing to catch the local aspect of the second definition:
Q2. How is the second definition local?
In particular, I'm struggling to understand the meaning of this example:
Example. Let $X:= \mathbb{V}(x_1x_4-x_2x_3)\subseteq \mathbb{A}^4_k$, the open subset $U:= X - \mathbb{V}(x_2,x_4)\subseteq X$. The function $$ (x_1,x_2,x_3,x_4) \longmapsto \begin{cases} \frac{x_1}{x_2} &\text{if $x_2\neq 0$},\\ \frac{x_3}{x_4} &\text{if $x_4\neq 0$} \end{cases} $$ is easily seen as regular on $U$ in the sense of the first definition, and has no representation as a quotient that works in all of $U$, which is fine by the definition.
The thing is, I can't wrap my head around using the second definition to study this example. I don't think the function is regular according to that definition. I don't even think is in $k(X)$.
Q3. What is going on? Is the function actually regular in the sense of the second definition? How is it an element of $k(X)$?
Thanks in advance.