You’ve given excellent explanations on this topic. It’s rare that someone writes about the meaning of the Incompleteness theorems correctly.

I think it’s worth pointing out or adding that when talking about The Natural Numbers most people are implicitly talking about the standard model with the first order Peano axioms. There are statements in this model that are true (have no counterexample) but which can’t be proven by this set of axioms.

I think many people making claims like,

“There are true statements about The Natural Numbers that aren’t provable.”

don’t realize exactly the nuances you’ve pointed out. Assuming The Natural Numbers are consistent then all true statements are provable in some axiom system. Just take the collection of all true statements as the axiom system. Now every true statement is trivially provable.

I hope what I’ve written doesn’t muddy your excellent explanations!