Mathematical proofs often, if not always, have some gap, and consequently, we have the 'convincing' part of the definition. Note that the author indicates gaps in his/her proof as he/she talks about the rule that from
$a=b$ and $b=c$ we can deduce $a=c$. I would take such basics of equational reasoning as assumed for some arithmetic, but I don't regard the partitioning of natural numbers into "even" and "odd" as something assumed for natural numbers. So, personally, I'm not so bothered by a proof getting rejected if it gets found as 'unconvincing'.
Consider the text that you cited where there existed a purported proof that if the square of a natural number is even, then the natural number is even. The author showed that the assumption of $n$ as odd leads to $n^2$ as odd. I have no contention with that, and that does show that such an assumption, along with the assumption of $n^2$ as even, leads to a contradiction. But, the author jumped to the conclusion that $n$ must be even. I don't find that convincing as a proof, because one might believe that there exists a third category of natural numbers other than odd or even, and some members of that category have even squares. Or someone might not have proved that such a partition exists for the arithmetic at hand. So, as a proof, I don't find it convincing, unless I missed some part of the text where the author first established that every natural number is exclusively either odd or even.
When I said that I don't find the proof convincing, I'm not just saying that the proof has not gotten formalized. When I said that I didn't find the proof convincing, I mean that were the proof to get formalized it would need non-logical notions not implied by the text in addition to what it already contains. Though I don't find the partitioning of natural numbers into odd and even numbers as by any means unachievable, it's not something that arises purely from logic or equational reasoning, but partly has to do with the nature of natural numbers. The reasoning of the proof may be regarded as valid in the sense that it draws a correct conclusion and no step moves from a true step to a false step, but the reasoning is inadequate in that it doesn't provide adequate details for deducing it's final claim. Specifically, it is correct to say that "if a natural number $n$ cannot be odd, then $n$ must be even", but that I don't think that consists of an axiom of the arithmetic of the natural numbers. So far as I can tell, the proof needs the concept of even and odd numbers as forming a partition of the natural numbers, and the author did not prove that, nor even spell it out as a needed lemma for the proof. Thus, the proof has a non-logical gap, and if someone did not see how that gap can get filled in, or believe that filling in that gap poses a serious challenge, rejecting the proof seems perfectly reasonable.
Also, sometimes mathematicians having different requirements for what constitutes a proof. Have you heard of intuitionistic logic or constructivist mathematics? Some mathematicians won't allow for a use of the law of the excluded middle in mathematical proofs. So, they can reject certain proofs.