Order statistics were introduced in my text as follows:
I am trying to understand what this means.
$X_1 , \dots , X_n$ is a random sample, i.e. an independent and identically distributed sequence of random variables.
$Y_1$ is the "smallest" of these $X_i$. Do I interpret this as $Y_1 = \min \{ X_1 , \dots , X_n \}$? Then $Y_2$ is the second smallest of the $X_i$ when realized, $Y_3$ the third smallest, and so on until $Y_n = \max \{ X_1 , \dots , X_n \}$?
An example to illustrate my thinking:
Suppose we perform an experiment and the random variables $X_i$ are realized as $x_i = i$ (so $x_1 = 1, x_2 = 2$ etc..). Then the random variables $Y_i$ are realized as $y_i = i$ as well because the $x_i$ are already in order.
If we perform another experiment and the $X_i$ are realized as $x_i = -i$ (so $x_1 = -1, x_2 = -2$ etc..), then this time the variables $Y_i$ were realized as $y_i = -n + i - 1$ (so $y_1 = -n, y_2 = -n + 1, \dots , y_n = -1$).
Is this the correction interpretation?
My confusion/hesitation comes from the fact that these $X_i, Y_i$ are random variables and therefore functions instead of values and it's not really clear from the text what is meant by ordering them.
Remember that, given a random sample $X_1,X_2,\dots,X_n$, we can talk about its sample mean $\bar{X}=\tfrac1n\sum X_i$. This "statistic" is random, just like the random sample itself. If the random sample is realized as specific real numbers $x_1,x_2,\dots,x_n$, then its sample mean will be realized, too, as a specific real number $\bar{x}=\tfrac1n\sum{x_i}$. Conceptually, it shouldn't be a big leap from the sample mean $\bar{X}$ to the sample minimum $Y_1$, or the sample maximum $Y_n$, or even the sample minimum-but-one $Y_2$, or more generally, the order statistics $Y_1,\dots,Y_n$. Like the sample mean, these statistics are random variables, and like the sample mean, given a particular realization of the data $x_1,\dots,x_n$, they will be realized as specific values $y_1,\dots,y_n$. It turns out that their realization will be a list of numbers that is the sorted version of the list of numbers in the realized data, but you should think of these statistics as evaluating functions of the data, not algorithmically re-ordering the data as some kind of non-functional "process" (see more on this below).
All of your observations in the rest of your question appear to be correct, and I don't see any problem with your interpretation.
With regard to your confusion/hesitation, as @DominikKutek pointed out in a comment, the informal intuition in terms of abstract "random quantities" can be formalized by considering the random variables describing the data as functions of a sample space $X_1(\omega),\dots,X_n(\omega)$. Then, the random variables describing, say, the sample mean or, say, the order statistics are also functions of the sample space: \begin{align*} \bar{X}(\omega)&=\frac1n\sum X_i(\omega)\\ Y_1(\omega)&=\min\{X_1(\omega),\dots,X_n(\omega)\}\\ &\dots\\ Y_n(\omega)&=\max\{X_1(\omega),\dots,X_n(\omega)\} \end{align*} The omitted definitions for the rest of the order statistics represent a notational problem, not a conceptual problem. We unfortunately don't have a convenient "name" for the mathematical "minimum but one" function $f\colon\mathbb{R}^n\to\mathbb{R}$ that we'd like to use in a definition: $$Y_2(\omega)=f(X_1(\omega),\dots,X_n(\omega))$$
Anyway, given a particular $\omega_0\in\Omega$, we can realize the data as real numbers $X_1(\omega_0),\dots,X_n(\omega_0)$, and the statistics above will be realized by evaluating them at $\omega=\omega_0$.
The definitions of $Y_1(\omega),\dots,Y_n(\omega)$ will have the "effect" of ordering the random variables, in the sense that the realized values at $\omega_0$ will be a sorted version of the realized data $X_1(\omega_0),\dots,X_n(\omega_0)$, but don't think of this ordering effect as some separate "process" that operates outside of the usual mathematical representation of random variables. It's all just functions of $\omega\in\Omega$.