I am familiar with the construction and general properties of trace fields for hyperbolic $3$-manifolds. But in that setting we use Moscow-Prasad rigidity to define these fields as manifold invariants. I want to understand how the trace field can be defined as a manifold invariant for a hyperbolic surface.
In particular, if $X$ is a complete, orientable, finite-volume, hyperbolic $3$-manifold, then by Mostow-Prasad Rigidity, the set of discrete faithful representations of $\pi_1(X)$ into $\mathrm{PSL}_2(\mathbb{C})$ makes up an isomorphism class of Kleinian groups. Since these isomorphisms are given by conjugation, which doesn't effect the trace, we can pick one of these Kleinian groups and use the traces in it to form the trace field as a manifold invariant.
Now let $S$ be a complete, orientable, finite-volume hyperbolic surface. Then $\pi_1(S)$ admits a Teichmüller space of discrete faithful representations into $\mathrm{PSL}_2(\mathbb{R})$ which are not necessarily isomorphic. How is the trace field defined, exactly, in this setting, as an invariant of $S$? Is it formed by all the traces from all the representations?
Also, a reference on this would be cool. I dug around for a while before posting this but could not find anything with enough detail.
You are not going to get very much information by studying the set of traces of all elements of $S$ over all discrete faithful representations of $S$.
For example, consider any nontrivial, nonperipheral element $\gamma \in \pi_1(S)$ represented by a simple closed curve. Using Fenchel-Nielsen coordinates, given any positive length $\ell > 0$ I can find a complete, finite area hyperbolic structure on $S$ in which $\gamma$ has length $\ell$. Thus, given any $T > 2$, I can find a discrete, faithful, finite area representation of $\pi_1(S)$ in $\text{PSL}_2(\mathbb{R})$ for which $|\text{trace}(\gamma)|=T$. The set of absolute values of traces of all elements of all representations is therefore equal to the entire interval $(2,+\infty)$.
Instead, what people study is the trace field of a particular point in Teichmüller space. That's an interesting and informative thing to do.