According to Wikipedia...
For example, suppose that four numbers are observed or recorded resulting in a sample of size $4$. If the sample values are $6, 9, 3, 8$, the order statistics would be denoted $x_{(1)} = 3,\ \ x_{(2)} = 6,\ \ x_{(3)} = 8, x_{(4)} = 9$, where the subscript $(i)$ enclosed in parentheses indicates the $i$th order statistic of the sample.
The first order statistic (or smallest order statistic) is always the minimum of the sample, that is, $X_{(1)} = \min\{X_1, ..., X_n\}$, where, following a common convention, we use upper-case letters to refer to random variables, and lower-case letters (as above) to refer to their actual observed values.
How is the transition made from $x_1$ to $X_1$? Ordering observed values $x_1, ..., x_n$ (i.e., the specific rolls of identically distributed or non-identically distributed dice) makes sense, but ordering random variables $X_1, ..., X_n$, (i.e., the dice themselves) should consist of stipulating a meaning to the notion of ordering the probability distributions associated with a collection of non-identically distributed random variables; as a programmer must stipulate what member variable or function thereof constitutes the order metric for a user defined class, one would choose between mean, median, or any other real-valued statistic of the random variable to be used for its ordering.
Why are random variables used in order statistics seemingly for the role that observed values should play?
My interpretation is that the random variable $X_{(i)}$ is defined as having a distribution corresponding to the $i$th smallest value in a set of $n$ iid samples.
You can't do the ordering until you have the actual sample values, but you can describe the distribution of values that the $i$th smallest sample will take.
Below is from the introduction of "Order Statistics and Inference" by Balakrishnan and Cohen. They are clear to make a distinction between the random variables and the values of the order statistics (note the non-standard notation), but it still corresponds with what's on Wikipedia.