I don't understand how Uspensky's definition is different from Eliezer's. Is there some minimum number of people a proof has to convince? Does it have to convince everyone? If I'm the only person in the world, is writing a proof impossible, or trivial? It seems that both definitions are saying that a proof will be considered valid by those people who find it absolutely convincing. And those people who do not find it absolutely convincing will not consider it valid. More importantly, it seems that this is all those two definitions are saying, which is why neither of them is very helpful if we want something more concrete than the colloquial sense of proof.
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.
If it's worth saying, but not worth its own post, even in Discussion, it goes here.