I am very new to this whole blockchain thing, so correct me if i am wrong, . . but someone claiming to be Satoshi Nakamoto could prove it by announcing a specific data transfer from the Satoshi Nakamoto account , (say 768.00987 BTC) and then after announcing such a claim, transfer that exact amount a few days later, . .the data transaction could be monitored for proof of ownership of that account at least. .
There is no such thing as a "Satoshi Nakamoto account".
There are public keys that have bitcoins associated with them, but it is impossible to know with 100% certainty which of those public keys are exclusively under the control of Satoshi Nakamoto.
Access to the private keys associated with the first few blocks, or the private key associated with the first bitcoin transaction would be a very strong indicator, but it isn't necessarily "proof" if Satoshi failed to properly secure those private keys from anyone else accessing them.