Should a mathematical proof be 'convincing'?

The primary reason to write down a proof is in order to communicate with other mathematicians. If mathematicians acquainted with the relevant literature cannot understand your argument (i.e: do not find it convincing) then I think the author has failed to write down a proof.

Everything about a mathematical proof is psychological. Including the valid and convincing part. Who gets to decide if it is valid and if it is convincing? The primary audience for a proof is the writer of the proof. Presumably the writer thinks it is valid and convincing. If the proof is convincing enough, then others may agree.

Understand that what is valid and convincing depends on time and place. The geometric proofs of ancient mathematicians were thought then to be valid and convincing enough at the time, but Later mathematicians have found that those proofs were not rigourous enough. For a nice discussion about this read the MSE question 80930 "What is the modern axiomatization of (Euclidean) plane geometry?".

It becomes a matter of communication and cooperation between the writer and reader of a proof. A proof by itself has no power to convince the reader who does not want to be convinced, no matter how rigorous or valid the proof may be. The proof may be rejected for several different reasons. There are many examples of this in mathematics history. A good example is the controversy over Cantor's theory of sets.

As for the last hypothetical question:

If there were only one single mathematician in the world, would s/he be able to produce a mathematical proof?

I think it is obvious that mathematician M could do whatever M wanted and nobody else could have mathematical objections. If M is convinced about the validity of a mathematical proof, then that is all that is required of a proof. After all, who decides if any given proof is "fallacious", if not mathematician M? Of course, M might detect a mistake in a proof and correct it. Thus, validity is time dependent. The entire idea of a "proof" is an artifact of history. Without context, what is the raison d'etre and meaning of writing and reading proofs?

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.

Suppose the definition excluded the word "convincing" and thus was:

A mathematical proof is a finite sequence of mathematical assertions which forms a valid argument for the desired conclusion from stated assumptions.

Now suppose you have a sequence of mathematical assertions that meets the definition above.

Now if another individual reads and agrees that your sequence of mathematical assertions is valid, and thus agrees it constitutes a proof by the above definition, then one could argue that you have created a "convincing" proof, and thus including or excluding "convincing" in the definition in this scenario would not matter.

But suppose instead however, you cannot find another individual who agrees that your sequence of mathematical assertions is valid. The key question here is then whether what constitutes a "valid sequence of assertions" is a matter of objective fact, or a matter of social agreement. If the latter, then it becomes impossible to have a "valid" yet "unconvincing" sequence of assertions (since then a sequence of assertions would only be "valid" if there is social agreement, which requires "convincing"). In this case, again, adding "convincing" to the definition would not affect whether a sequence of assertions is classified as a proof or not. However if the validity of an argument is an objective property that exists independently of people, then one could argue that an unconvincing, yet valid argument could indeed exist, which I believe would merit being labelled a "proof".

Therefore I have shown that depending on your philosophy of mathematics either (1) adding "convincing" to the definition doesn't change the content of the definition or (2) a proof could exist that is not "convincing". As such, whether (1) or (2) is adopted, the word "convincing can be safely removed from the definition of a proof.