big picture - Defining variable, symbol, indeterminate and parameter

In written English (and of course other languages), we have linguistic constructs which tell the reader how to approach the ideas that are about to be presented. For example, if I begin a sentence with "However, . . .", the reader expects a caution about a previously stated proposition, but if I begin the sentence with "Indeed, . . . ", the reader expects supporting evidence for a position. Of course we could completely discard such language and the same ideas would be communicated, but at much greater effort. I regard the words "variable", "constant", "parameter", and so on, in much the same way I regard "however", "indeed", and "of course"; these words are informing me about potential ways to envision the objects I am learning about. For example, when I read that "$x$ is a variable", I regard $x$ as able to engage in movement; it can float about the set it is defined upon. But if $c$ is an element of the same set, I regard it as nailed down; "for each" is the appropriate quantifier for the letter $c$. And when (say) $xi$ is a parameter, then I envision an uncountable set of objects generated by $xi$, but $xi$ itself cannot engage in movement. Finally, when an object is referred to as a symbol, then I regard its ontological status as in doubt until further proof is given. Such as: "Let the symbol '$Lv$' denote the limit of the sequence $lbrace L_{n}v rbrace_{n=1}^{infty}$ for each $v in V$. With this definition, we can regard $L$ as a function defined on $V$. . . "

So in short, I regard constructing precise mathematical definitions for these terms as equivalent to getting everyone to have the same mental visions of abstract objects.