I'm just looking for some clarification about the concept of free variables, and their role in the definition of sets definable in a structure. Sorry if it's a little open-ended, though any help is appreciated.
I've notice that various logic books have different conventions for notation when presenting the free variables that (may) reside in a formula. I'll just quickly establish notation: Let $\sigma$ be a signature for a language of first order logic, and let $V$ be the countable set of all variables. Let $\Phi_\sigma$ denote the set of all formulas over this signature. For $\phi \in \Phi_{\sigma}$, let $F_{\phi}$ denote the set of all variables $x \in V$ that occur freely in $\phi$. Let $\mathbb{A}$ be a $\sigma$-structure, with domain $A$.
Now, some books will differ on what it means when we say $\phi(\vec x)$, where $\vec x = (x_0, \dotsc, x_{n-1})$ is a sequence of variables in $V$ of length $n$, with the intent of determining, given $\vec a \in A ^ n$, whether $\mathbb{A} \models \phi(\vec a)$. For example, some will consider $\vec x$ to consist of exactly the free variables that occur in $\phi$, with each occurring once. In other words, that it's an enumeration of $F_\phi$. Others will relax this and say that when we write $\phi(\vec x)$ out of context, we are implicitly assuming only that $F_{\phi} \subseteq \{x_{0}, \dotsc, x_{n-1}\}$. Others will not require this at all, and allow not only more variables than occur freely in $\phi$, but also not require that the sequence contain all of them. Thus, in this latter convention, the choice of $\vec x$ is somewhat arbitrary. In the first convention above, one can at least require a well ordering on $V$, and say, given $\phi$ above, there is a canonical increasing enumeration $\vec x _{\phi}$ of $F_{\phi}$.
In a given structure $\mathbb{A}$, and given a formula $\phi \in \Phi_\sigma$, one can define the set/relation in $\mathbb{A}$ defined by $\phi$ to be $D(\phi) := \{\vec a \in A ^ n : \mathbb{A} \models \phi(\vec{a})\}$. However, this would depend then on the sequence $\vec x$ chosen for this, right?
Considering for now the lattermost convention above, would it not matter, even for the same formula $\phi$? If we consider distinct $\vec x = (x_0, \dotsc, x_{n-1})$ and $\vec y = (y_0, \dotsc, y_{m-1})$ then we can get different sets, even of different dimensions, so I'm imagining that this chosen sequence should be viewed as a parameter in a definable set (as say we should write $D_{\vec x}(\phi)$ instead). In fact, there is technically no requirement that the sequences $\vec x$ even need be injective, so that there could be repeats. It would at least make sense from a technical standpoint since it would just result in a repeat of coordinates. The same goes for occurrences $x_i$ that don't occur freely in $\phi$, or occurrences of variables that don't appear among the $\{x_i\}_{i \in n}$, since they still make sense from a technical standpoint to ask if $\mathbb{A} \models \phi(\vec x)$, even if silly.
Is it the case that a set definable by a formula $\phi$ with a presentation of variables $\vec x$ with certain silly features (such as with repeated, missing, or extra variables), could also be definable by a modified yet 'similar' formula without them? If so, I guess we could talk about definable sets in a matter that, without loss of generality, avoids that and is more like the former most convention, with the canonical 'sensible' choice of variables.