Ownership is only confirmed by outgoing transactions, not by incoming transactions, so if Craig Wright wants to prove he is Satoshi Nakamoto, he needs to send at least 1 satoshi from Satoshi address to a specified address. Otherwise, all other attempts at proof are in vain.
It is true but even not enough.
We can say "Not your keys (private keys), not your bitcoins" but if someone can have access to a private key, send bitcoins, it does not mean the person is an actual original owner of the wallet.
The person can be someone else, just get access to the private key.
So if Craig Wright wants to prove ownership, he must do at least three things.
Move bitcoins from Genesis address.
Sign a Bitcoin message by that a private key of Genesis address.
Sign a PGP message from Satoshi Nakamoto PGP key.
About making a transaction with 1 satoshi, it is not correct because with Bitcoin on-chain transactions, there are limits (minimum) for transactions, to avoid dust attacks.
Depends on address and transaction types, the minimum is different but it is about 330 - 546 satoshis at least.
Legacy addresses: 546 satoshi
Native Segwit addresses: 294 satoshi
Taproot addresses: 330 satoshi
How was the dust limit of 546 satoshis was chosen? Why not 550 satoshis?What is meant by Bitcoin dust?dust = (input_vsize + output_size) × 3 sat/vB