1 - Context
1.1 - (Relative) Open and Closed Sets
In Baby Rudin, we have the following definitions:
(Open Set) A subset of a metric space, $E\subseteq X$, is said to be "open" if for all $p\in E$ there exists some $r>0$ such that $N_{r}(p)\subseteq E$. (Note that in Baby Rudin, $N_{r}(p):=\{ x\in X \,|\,d(p,x)<r \}$)
(Closed Set) A subset of a metric space, $E\subseteq X$, is said to be "closed" if for every $p\in X$, if $p$ is a limit point of $E$ then $p\in E$.
Then it is shown that whether or not a given set is open or closed depends on what metric space the set is embedded in. For example, $(0,1)$ is open in $\mathbb{R}$, but not in $\mathbb{R}^2$. Similarly, the set $(0,1]$ is closed in $\mathbb{R}^+$ but not in $\mathbb{R}$.
This means the concepts of "open" and "closed" only really make sense relative to some specified space. So we introduce the notions of relative open-ness and closed-ness:
(Relatively Open) Let $E\subseteq Y \subseteq X$. The set $E$ is said to be "open relative to $Y$" if for all $p\in E$ there exists some $r>0$ such that $Y\cap N_{r}(p) \subseteq E$.
(Relatively Closed) Let $E\subseteq Y \subseteq X$. The set $E$ is said to be "closed relative to $Y$" if for all $p\in Y$, if $p$ is a limit point of $E$, then $p\in E$. (Note that Rudin doesn't explicitly define relative closed-ness, this is just my interpretation of what he means by it, please correct me if I am wrong)
Then, it is shown that if $E\subseteq Y \subseteq X$, then $E$ is open relative to $Y$ if and only if there exists some set $G$ which is open relative to $X$ such that $E=Y\cap G$.
To interpret this result in terms of our previous example, for $(0,1)$, $G$ could be $N_{0.5}(0.5)$, and for our example of $(0,1]$, $G$ could be $[0,1]$.
1.2 - Open Coverings and Compact sets
Okay, after talking about relative open-ness/closed-ness, Rudin introduces the notion of an open covering, and a compact set.
(Open covering) By an "open cover" of a set $E$ in a metric space $X$ we mean a collection $\{ G_{\alpha} \}$ of open subsets of $X$ such that $E\subseteq \bigcup_{\alpha}G_{\alpha}$.
(2.32)(Compact set) A subset $K$ of a metric space $X$ is said to be "compact" if every open cover of $K$ contains a finite subcover.
Then Rudin remarks that unlike how the open-ness and closed-ness of a set may change depending on the space the set is embedded in (e.g., the examples above), the compact-ness of a set does not change depending on the space the set is embedded in.
To formally state (and prove) this fact, Rudin introduces the notion of "relative compact-ness". Specifically, he says that a set $K$ is "compact relative to $X$" if the requirements of definition 2.32 are met. This is where my confusion arises.
2 - Question/Confusion
2.1 - Nonstandard Definitions
To distinguish between some of the different ways that I think the meaning of "$K$ is relatively compact to $Y$" can be interpreted, I am going to create some definitions. I do not know if these definitions have pre-existing standard names, I am just making them up so that I can be clear with how I formulate my question.
($X$-Cover) Given a metric space $X$, and a set $K\subseteq X$, we define the "$X$-cover of $K$" to be a collection $\{ G_{\alpha} \}$ of subsets of $X$ such that $K\subseteq\bigcup_{\alpha}G_{\alpha}$.
(Open $X$-Cover) Given a metric space $X$, a set $K\subseteq X$, and an $X$-cover of $K$, say $\{ G_{\alpha} \}$, we say that $\{ G_{\alpha} \}$ is a "Open $X$-cover of $K$" if each $G_{\alpha}$ is open relative to $X$.
(Relatively Open $X$-Cover) Given a metric space $X$, a set $K\subseteq X$, an $X$-cover of $K$ (say $\{ G_{\alpha} \}$), and a metric space $Y$ such that $K\subseteq Y \subseteq X$, we say that $\{ G_{\alpha} \}$ is a "Open $X$-cover of $K$ relative to $Y$" if each $G_{\alpha}$ is open relative to $Y$.
So the main thing that I am trying to describe with these definitions is that if we have $K\subseteq Y \subseteq X$, an Open $Y$-cover of $K$ (say $\{ F_{\beta} \}$), and an Open $X$-cover of $K$ relative to $Y$ (say $\{ G_{\alpha} \}$), then all of the $F_{\beta}$ are subsets of $Y$ (and are relatively open to $Y$). But it may not be the case that all of the $G_{\alpha}$ are subsets of $Y$, we only require that they are relatively open in $Y$.
2.2 - Question
So my problem is I believe there are two distinct natural interpretations of "relatively compact", and I want to know which of these is the correct one (or at least the one Rudin means). If neither of these are correct, I would be seriously grateful if someone could help guide me to understanding! Ok, here are what I think the two possible interpretations are, using the definitions I have made-up just to be precise:
(Interpretation 1) (Relatively Compact) Given $K\subseteq Y\subseteq X$, we say that $K$ is "relatively compact to $Y$" if every Open $Y$-cover of $K$ has a finite subcover.
(Interpretation 2) (Relatively Compact) Given $K\subseteq Y\subseteq X$, we say that $K$ is "relatively compact to $Y$" if every Open $X$-cover of $K$ relative to $Y$ has a finite subcover.
Could someone please let me know which interpretation Rudin means? I am new to analysis and reading Rudin for the first time, and it is difficult sifting through all the definitions. I just want to make sure I am not misunderstanding something fundamental before moving onwards.
Your definition of "open $X$-cover relative to $Y$" says that the sets in the $X$-cover are open relative to $Y$, but this is not a well-defined notion, since $Y\subseteq X$, not vice versa. It would only make sense in the reverse direction - a covering of $K$ by sets in $Y$ could be defined to be "an open cover relative to $X$" if each set was open relative to $X$.
I will assume hereon out that that is what you meant.
In this case, there are in fact three possible interpretations I can see under your definition:
Interpretations 1 and 2 are equivalent and are the standard definition. To see they are equivalent, note that a set in $Y$ is open relative to $Y$ if and only if it is of the form $Y\cap U$ for some $U$ open relative to $X$.
Interpretation 3 is a much weaker definition and will often be vacuously true, as there are often no nonempty subsets of $Y$ that are open relative to $X$, meaning there are no open $Y$-covers relative to $X$ for any nonempty subset $K\subseteq Y$ (For a concrete example, consider $X=\mathbb R^2$, $Y\subseteq X$ a line). I don't know of any use of Interpretation 3 under any name.
Remark
As discussed in the comments, the term "relatively compact" generally refers to a different concept than Rudin's notion here.