This is in response to the new edited version of the question.
You are using "indicator" functions on $X$ but with respect of membership in $X$ instead of subset-hood of $X$.
This is more complex, your $X$ is what we usually think of as a domain of a model, your relation $E$ is the membership relation of the model which is defined on the domain, and your $\in$ is the element-hood in the domain, which is external to what the axioms are speaking about. Possibly this approach can work, but what's the point of it really. I mean why not take the simpler way of saying that we have a collection $X$ and stipulate ordered pairing as a primitive, axiomatize $\forall a,b \in X (\langle a,b \rangle \in X)$ and of course axiomatize the basic property of ordered pairs, then let $E$ be a collection of ordered pairs in $X$, then Define the atomic formula $x \ E \ y$
$$ x \ E \ y \iff \langle x,y \rangle \in E$$
Then write the axioms in terms of atomic formulas using $E$ with all there quantifiers bounded by $X$. Of course 'sets' are defined simply as 'elements of $X$'.
Those axioms would serve to lay down the basis for characterization of $E$.
To me that's simpler than taking indicator functions on the whole domain respective to elements in that domain, those functions would be outside of the domain itself, so how for example you'll quantify over those functions (you call as sets)? If you quantify over them then you enter second order logic arena? If you wont quantify over them, then you may us a constant logical pairing function of your, and possibly another constant one place function symbol $h(a): X \to \{0,1\}$ for an $a \in X$, then you present the axioms quantified over elements of $X$, and write down formulas in terms of $h$ and $E$, not that easy but it can be done I think. You need to have ordered pairs as primitives, symbols 0,1 as constants, also $\in$ and favorably $=$ as primitives. It can be done I suppose, but I don't know what is the point behind this? It appears more complex to me.