"AXIOM: A proposition regarded as self-evidently true,
Axioms being self-evidently true means that if you choose to DENY them, you need to USE them in the denial process anyway, therefore proving that they ARE true. Try assuming that the so called identity principle is not 'true' and see where you get.
No, one simply denies them. But, even if one HAD to USE them, that would not prove them. I might use several axioms to derive a contradiction. If one of my axioms IS the law of non-contradiction, I can then exclude one, or more of the axioms to rectify the situation, including the principle of non-contradiction. I'm not aware of any systems of logic that deny the law of identity. It is, of course, quite foundational. But since you claimed that axioms can be proven: well go ahead and prove it, then. After all, the burden of proof is never on the skeptic. But this all began not with a disagreement over the law of identity, but over the law of non-contradiction. The principle of non-contradiction IS denied, by example, in the philosophical school of dialetheism. It is used in mathematics, as well, in formal paraconsistent logics, which can not only be inconsistency-tolerant, but also have more states to a proposition than merely "true" and "false." These, and other multi-value logics, generally, are useful in a wide area of mathematics, physics, electronics and so on. In these complex systems they are quite useful. Why should we not use them in other complex areas of thought? Simply because you're not aware of them? Well then, OK. So then we have to AGREE on which system to use first, then. "QED"
I don't need to invoke internet authority or the sacred wikipedia scriptures xorcist. I can explain what an axiom in my own words.
I don't either. I've explained several times that axioms are assumptions, or if you prefer, propositions, and are not subjected to PROOF. You disagreed, so I quoted sources. Your statement about axioms was quite wrong. They are not subject to proof. You're now trying to walk that back and play a different angle -- one that I handed to you -- that of self-evidence. Quite good. Let's continue.
And even going by that definition, if axioms are self-evidently TRUE, then people who don't 'agree' that they are true are self-evidently...troubled.
They are self-evident only from a certain way of thinking. For example, in geometry, a point is axiomatic. An infinitely small object with no dimension. This is an intellectual abstraction that is self-evident and meaningful only from the perspective of that intellectual simplification used in geometry. Physically, there is nothing with no dimension. From an idealized, abstracted, way of thinking the existence and meaning of a point is self-evident. From a physical perspective, it is utter rubbish. What perspective you have, the way you're approaching a subject, informs which axioms you'll use as the foundation of your reasoning.
(and by the way, your first source is about mathematics, not logic or philosophy in general)
I'm quite aware of that. So? Formal logic, as studied and used by philosophy, is a branch of mathematics. And in any case, in philosophy, and informal logic, axioms cannot be proven, as you have stated.