If a 'variable assignment' function maps from a set of symbols, would it be correct to formulate a variable as simply a particular symbol that performs the role of a variable in my language? So when we write "'x' is a variable" we are unambiguously referring to the symbol 'x' which is indeed a variable in our language?
In a formally-defined language, there is a list of defined symbols. The language may classify symbols according to their meaning. Common classes include constants, functions symbols, predicate symbols, connectives, quantifiers, and variables. So, yes, a variable is a symbol.
What makes a symbol a variable (other than the stipulation of the language)? There are basically three ways that a symbol is given a meaning in formal languages:
An interpretation rule which directly specifies what a symbol means. For example, one rule might be that [P & Q] is true if and only if P is true and Q is true. This defines how the symbol "&" is to be understood in the language.
A mapping in an interpretation function. An interpretation function maps symbols in the language to some set of values which we might call the domain. For example, the symbol "0" might be mapped to the number 0, and the symbol "succ" might be mapped to the successor function.
A variable assignment, which, like an interpretation function is a mapping from symbols to the domain. But unlike an interpretation function, a variable assignment varies with context. For example there might be an interpretation rule that [∀xP] is true if and only if P is true for every variable assignment that assigns x to any member of the domain.
A variable assignment is like an interpretation because it maps symbols (specifically variables) to values in the domain, but the interpretation function is the same everywhere, while a variable assignment varies with context as defined by an interpretation rule. Also there isn't typically a single variable assignment that applies to a variable in a sentence. Normally, there is a set of variable assignments as in the example of [∀xP].