Judging from the AR quote you posted on axioms, Ayn Rand's definition of axiom is no different than the definition of absolute - are these synonyms? The dictionary defines axiom as "a universally recognized truth" however, there is no such thing as a "universally recognized truth" in human-land because people disagree about everything. In effect, axioms are "truth by definition." From your post, Ayn says that "axiomatic concepts refer to facts of reality and are not a matter of "faith"", however, how does a human decide which principles are axioms and which aren't? The only principles that one can authoritatively say are universal are the ones that come from God. Any other is a human invention and does not have universal applicability, regardless of whether someone labels them as "axiomatic." Of course, that would mean that many of AR's philosophical beliefs are her opinion only which she calls axioms, but insofar as they disagree with God's moral absolutes, they are certainly not universal truths.
I will assume your question is sincere. I have never seen this explicit statement by Ayn Rand. (She does say the attempt to prove axiomatic concepts is self-contradictory, by which she means you have to start with a contradiction to make the attempt, that is, you have to assume what is true to be not true.)
Suppose you oberve something about reality or some aspect of it, and identify it. It might occur to you that the observed aspect of reality seems almost fundamental, essential, or primary. Is it an axiom?
Here is how to find out. Is it possible to deny it without incorporating the idea itself in the denial thus contradicting the denial in the process of denying it. If the concept itself must be incorporated or assumed in the process of the denial, then it cannot be logically denied, it is axiomatic.
One other note about axioms. Axioms are always primaries, which means, they cannot be further reduced to more fundamental concepts. They are not derivative concepts.
That is what objectivists mean by axioms.
For the record, Randian Objectivists (and I think those in the Kelly camp as well) hold that there are only three axioms: existense, consciousness, and identity. In axiomatic form they would be, Existence exists. Conscious is conscious. A is A, or a thing is what it is.
Since I am not an objectivist, I am not bound by their limitations, and take the risk of thinking for myself. One conclusion is that there are more axioms.
For example:
Plurality. (There must be more than one thing.)
Dynamism. (There must be change.)
Differentiation. (All existents are different.)
And others, on the basis that once these are observed, they cannot be denied without entailing a self contradiction. But don't hold the objectivists to account for these.
Hank