Before going into the topic first let me clarify what I will refer to as "god" in this passage. God, I think should be referred to as something that is analogous to the axioms of a formal system. First of all, we have not been able to define god, and possibly cannot, since we are within the system. Verification of existential claims can be done in two ways. One is the empirical verification. Suppose you are told that a centaur exists. You know all the properties of the creature. In that case you can go in search of the creature yourself. This is empirical verification. The other way is to run an ontological argument using the definition of the object the existential claim of which you have to verify. For example, if you are told that a four-sided pentagon exists, you need not go in search of one, since by definition pentagons are five-sided. In this case you are basically using the definition to arrive at a contradiction. Many philosophers have tried to define god in many ways. In every definition I have seen, there is some inconsistency. My point is that since god (if it exists) is analogous to the axioms of a formal system, how can we define god within our system? And if we can't define god precisely then how can we prove or disprove the existence of god within our system. Doesn't the question of existence of god seem to be logically not sound?
By invoking terms like "axiom" and "formal system," I naturally am inclined to answer from a mathematical perspective. What you describe calls for a self-referential system, so that it may refer to its axioms as elements of the system. The closest you can get to an existence argument using this analogy would probably be to assert the truthfulness of the axioms for God using the formal system itself.
Doing this puts you square in the realm of nuanced mathematical challenges like Godel's Incompleteness Theorem and Tarski's undefinability theorem, which put interesting limits on what can and cannot be said within that formal system. On the other hand, there's work like the work of Dan Willard which explore carefully crafted formal systems which skirt these theorems and manage to be "self verifying," meaning the system can be used to prove the systems validity.
Of course, all of this is making the assumption that reality is sufficiently analogous to a formal system to make these analogies worthwhile. A proof of that claim would be something indeed.
I, too, will take the mathematical route. According to https://en.wikipedia.org/wiki/Axiom:
As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. As modern mathematics admits multiple, equally "true" systems of logic, precisely the same thing must be said for logical axioms - they both define and are specific to the particular system of logic that is being invoked. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.
In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems. However, an axiom in one system may be a theorem in another, and vice versa.
Let's take logical axioms first. What would be the value in defining "god" to be the set of (true, self-evident) axioms within the system those axioms define? Yes, doing so would enable to one to answer the question "What is god?" but the answer seems sterile and unlikely to appeal to many.
With regard to non-logical axioms, such as the axioms of arithmetic, the axioms are merely starting points for the deduction of additional statements or sentences. Of course, one could define "god" as this set of axioms, but, again, what would be the value? The same questions some people raise about "god", they could now raise about the axioms, e.g., Are the axioms "true"? Are the axioms "consistent"? Are the axioms "complete"? It's not obvious what one would gain by this.
A few comments on Goedel. Among other things, he's showed that:
There are an endless number of true arithmetical statements which cannot be formally deduced from any given set of axioms by a closed set of rules of inference. (Goedel's Proof, Nagel and Newman, New York University Press, 1958, p.98)
This suggests that rather than equating "god" with the axioms that define a given mathematical system -- and which are part of that system -- it may make more sense to equate "god" with the set of all true statements that cannot be deduced from those axioms -- which are not part of that system.
All this presumes though, the validity of the mathematical analogy to begin with. On that and other matters, I agree with @CortAmmon.