Those are statements that fall under what Godel proved, that is they are statements that are unprovable in ZF. So even though his statement doesn't include self-reference it can still fall under Godel's proof if his decoder is strong enough to determine what is an integer and what is not an integer. Self-referencing has nothing to do with it at all.

The existence of specific undecidable statements in ZF or ZF - AC is a different sort of result than what Godel showed. That for example the continuum hypothesis is undecidable in ZFC is interesting because the continuum hypothesis is interesting. However, Godel's theorems show that any consistent, axiomatizable systemn with all valid proofs recursively enumerable, that is strong enough to model a large chunk of N, must be incomplete. Exhibiting results like Cohen's results about choice and the continuum hypothesis don't give you the full result, they just show specific things about the system ZF. Indeed, if one didn't know Godel's theorems, and just knew about Cohen's forcing results, you might be tempted to just throw in AC and GH as additional axioms if you didn't have them already and then wonder if that system is complete.

It is very funny to me that my most downvoted comment isn't about religion but is about Godel's proof and no one gave a decent refutation of what I said.

People here have a very complicated set of attitudes. Even as many don't want to bother talking about irrational aspects of religion, or engaging in religious individuals who are convinced that their religious view is somehow different from all the other religious views, people who are religious are that way due to a variety of very strong cognitive biases. So there's some sympathy there. Getting math wrong and then insisting one is correct is just simple arrogance or at least, only Dunning-Kruger. There's much less sympathy there. And yes, people have tried to explain why you were wrong albeit fairly succinctly.