You're right that it doesn't matter what we use as symbols, as long as they are mathematical objects that we can compare for equality, put into sets, etc.
There are at least two reasons for writing $c_x$ instead of just $x$ in situations such as the one you quote:
It reminds the human reader that the new symbols are just symbols: the proof systems and rules of logic don't care about their properties as numbers, except to the extent we define new axioms to force them to.
It allows us to rename the symbols if we need to -- for example it might be that some of the symbols that are already in the language happen to be numbers. (Usually we haven't said what exactly they are either, which means that we cannot be sure they are not numbers). In that case we need to choose something different for our new symbols.
So the content of what you quote is just
Choose some set $S$ of the right cardinality that is disjoint from the set of symbols we're already working with, and fix a bijection $c:\mathbb R\to S$. We will write $c(x)$ as $c_x$. Consider a new logical language whose symbols are our old vocabulary together with $S$ ...
The phrasing in your quote is intended to be the "minimally invasive" way to imply something like this, which is still explicit enough to avoid being sidetracked by clever students protesting, "but what if $2$ is already a symbol? Where did we assume it is not?"