As I understand Eliezer's definition: "Your text is proof if it can convince me that the conclusion is true"
As I understand Uspenskiy's definition: "Your text is proof if it can convince me that the conclusion is true and is I am willing to reuse this text to convince other people".
The difference is whether the mere text convinces me that I myself can also use it succesfully. Of course this has to rely on social norms for convincing arguments in some way.
Disclosure: I have heard the second definition from Uspenskiy first-person, and I have never seen Eliezer in person.
Does Uspenskiy have an opinion on Zero-knowledge proofs? They differ from standard proofs in that they have a probability of being wrong (which can be as small as you want), and the key property which is that if I use one to convince you of something, you aren't able to use it to convince anyone else.
If it's worth saying, but not worth its own post, even in Discussion, it goes here.