As I understand it, a physical theory that has a gauge symmetry is one that has redundant degrees of freedom in its description, and as such, is invariant under a continuous group of (in general) local transformations, so-called gauge transformations.
With this in mind, consider electromagnetism as a prototypical example. This has a $U(1)$ gauge symmetry, such that the theory is invariant under transformations of the vector four-potential, $A^{\mu}$ $$A^{\mu}\rightarrow A'^{\mu}=A^{\mu}+\partial^{\mu}\Lambda(x)$$ where $\Lambda(x)$ is some local function of space-time coordinates. Is it then correct to say that the theory is described by an equivalence class of vector four-potentials, $A^{\mu}$ such that $$A'^{\mu}\cong A^{\mu}\iff A'^{\mu}=A^{\mu}+\partial^{\mu}\Lambda(x)$$ Given this, does "choosing a gauge " simply amount to choosing a particular four-potential $A^{\mu}$ from this equivalence class?
Furthermore, does "fixing a gauge" simply amount to specifying some constraint on the choice of $A^{\mu}$ such that it "picks out" a single four-potential $A^{\mu}$ from this equivalence class?
For example, is it correct to say that choosing the Lorenz gauge $\partial_{\mu}A^{\mu}=0$ partially removes gauge freedom, since it restricts ones choice of four-potential $A^{\mu}$ such that it satisfies $\partial_{\mu}A^{\mu}=0$, however, it doesn't fully fix ones choice of $A^{\mu}$, i.e. it doesn't fully "fix the gauge", since there remains a subspace of gauge transformations that preserve this constraint, corresponding to gauge functions $\psi$ that satisfy the wave equation $\partial_{t}^{2}\psi=c^{2}\nabla^{2}\psi$?!
