I think “proof” as used in the post was meant to refer to neither the colloquial use nor fully-formalized statements, but proofs as they are actually used by human mathematicians: written arguments which the community of mathematicians believe could be formalized if necessary. Then the question is how much confidence we should have in a statement given that such a “proof” exists. Mathematicians are quite good at determining which proofs are valid, but as the post points out, they are not infallible.
I think “proof” as used in the post was meant to refer to neither the colloquial use nor fully-formalized statements, but proofs as they are actually used by human mathematicians: written arguments which the community of mathematicians believe could be formalized if necessary. Then the question is how much confidence we should have in a statement given that such a “proof” exists. Mathematicians are quite good at determining which proofs are valid, but as the post points out, they are not infallible.