From my understanding, signing a message and including it on the blockchain is one of the simplest ways to demonstrate control over a private key.
That is like a "double proof" because you can just spend coins from that address and the signature you produce in that transaction is just like the signature you'd use when signing a message. There is no need to sign a message and inject that into your tx.
Yes, its like "double proof". But, if you don't add some arbitrary data to your transaction, anyone could later claim to have made that transaction (using the same private key).
That's a good point. In that case a specific message could be included in the OP_RETURN output of that transaction. Such message is much smaller than including a signature+message in the transaction.