The theory is that there are statements that are true but undecidable with any sufficiently strong decider. Those statements are proofs of that theory, they (or their negation) are true but undecidable.
You are right, in that those statements do satisfy the conclusion, and I don't see why you're being downvoted. The difference between them and the original, self referential, statement is generality. For example:
You: I know that ZF set theory is incomplete because the axiom of choice cannot be proven within it.
Some other guy: Okay then, how about we add the axiom of choice as another axiom, maybe this new system will be complete?
You: Nope, it still can't prove or disprove the continuum hypothesis.
SUG: So I'll add that in as another axiom, maybe that will f...
Followup to: What's a "natural number"?
While thinking about how to make machines understand the concept of "integers", I accidentally derived a tiny little math result that I haven't seen before. Not sure if it'll be helpful to anyone, but here goes:
You're allowed to invent an arbitrary scheme for encoding integers as strings of bits. Whatever encoding you invent, I can give you an infinite input stream of bits that will make your decoder hang and never give a definite answer like "yes, this is an integer with such-and-such value" or "no, this isn't a valid encoding of any integer".
To clarify, let's work through an example. Consider an unary encoding: 0 is 0, 1 is 10, 2 is 110, 3 is 1110, etc. In this case, if we feed the decoder an infinite sequence of 1's, it will remain forever undecided as to the integer's value. The result says we can find such pathological inputs for any other encoding system, not just unary.
The proof is obvious. (If it isn't obvious to you, work it out!) But it seems to strike at the heart of the issue why we can't naively explain to computers what a "standard integer" is, what a "terminating computation" is, etc. Namely, if you try to define an integer as some observable interface (get first bit, get last bit, get CRC, etc.), then you inevitably invite some "nonstandard integers" into your system.
This idea must be already well-known and have some standard name, any pointers would be welcome!