I have a Danish book about the theory of knowledge for mathematicians which I have tried my best to translate some parts into English.
According to the lecturer, we can with "certain reasonability" call the understanding of proof with three points as shown below as Standard View of Proof,
1) A proof must be linearly (that is, there should be a unique way through the chain of deductions from axioms to the proven theorem).
Danish: Et bevis skal være lineært i den forstand, at der skal være en entydig vej gennem kæden af deduktioner fra aksiomer til den beviste sætning.
2) A proof must be propositionally (that is, all claims and conclusions should be explicitely expressed statement in a (ie. formally) language that is already established).
Danish: Et bevis skal være propositionelt i den forstand, at alle påstande og slutninger skal være eksplicit formulerede udsagn i et (evt. formelt) sprog, som er etableret i forvejen.
3) A proof must be formalizable (that is, it should be possible to express the proof like a chain of deductions in a formally system).
Danish: Et bevis skal være formaliserbart i den forstand, at det skal være muligt at formulere beviset som en kæde af deduktioner i et formelt system.
I have been thinking about their concepts for a while and I still am unsure about it when comparing it with any proofs. I would like to use a proof like a example (say Cauchy–Schwarz inequality in LINK) in order to gain the understanding. I would say that the proof in the link satisfies 2) and 3) since we are convicted it to be true.
I think you shouldn't put too much weight on that description. It doesn't sound like something a mathematical logician would write, and it certainly doesn't do a very good job of capturing my impression of how proofs are usually viewed.
(1) is just plain wrong. More often than not, actual proofs have an inherently branching structure, for example proofs by case analysis. When we write down the proof we put one of the branches before the other, but that's just a bow to the linear nature of our communication methods -- it's not connected to the deeper "proofiness" of the proof.
As for (2), it is right that claims that one considers during a proof are expected to have clear criteria for being true or false. But again, that is not really particular to proofs. We make the same demand of definitions, or to the statement of theorems (as distinct from their proofs). Really, all of technical mathematical prose is expected to live up to this standard -- except for parts explicitly declaring themselves to be about fuzzy intuitive analogies.
(3) is on one hand true -- people do expect proofs to be formalizable -- but on the other hand it also has a smell of putting the cart before the horse. During most of the history of mathematics (all the way up to about 1900, give or take several decades) the concept of formalizing proofs didn't exist yet, but mathematicians proved things all the same, and from around 1800 the overall structure and style of their proofs were pretty much the same as proofs written today.
A general expectation for proofs to be formalizable only arose after suitable formalizations were invented, and after a generation or two of debate resulted in a consensus that certain formal systems in fact do capture (almost) everything that we want to call a proof and nothing that we don't want to accept as a proof. The class of arguments one intuitively wants to accept comes first; the formalization comes later. It is interesting and fortunate that we have formalizations that work so well (not least because looking to the formal system is more civilized than having a shouting match if doubt about the validity of some reasoning step arises), but the reason they work is because they happen to give us what we agree to want anyway.
In short, don't sweat it if the points you quote seem to you to be somewhat removed from how proofs work in real life. You're probably even right. Instead read on, see if the author uses those characteristics to say something interesting. For example it sounds vaguely like they're being presented in order to contrast them with some other view of proofs later -- and knowing about such other views might well be interesting and useful even if the book doesn't do a particularly good job of articulating the mainstream view they contrast to.