Even if Satoshi presents his privatekeys, signs messages, shows his passwords, e-mail account accesses... it could be someone else who has all these privatekeys, passwords, accesses now.
He would only prove the ownership of these but it is impossible for somebody to prove that he is Satoshi.
Demonstrating the ownership of private keys is a very strong start - there are some possibilities how other people could have obtained them, but still it's the strongest proof. After that, we can look at what kind of a person the alleged Satoshi is - do they have some background in cryptography and programming, were they active in the Cypherpunk community and other similar movements, do they generally demonstrate a very strong understanding of Bitcoin and cryptography?
These things are very important too, if Craig Wright showed a valid Satoshi's signature tomorrow, I still wouldn't believe him and would think that he somehow stolen Satoshi's keys, because CSW consistently demonstrated that he's not a cypherpunk and just a conman.