Is there a way to prove ownership of a transaction? Is it possible to prove that you are the sender in a transaction?
Prove this to 3rd-parties, not only the receiver.
You cannot "own" a transaction. But you can own the private keys for one or more of the addresses from the inputs of the transaction.

A (regular) transaction must be signed with the private keys of the addresses of all the inputs. So if you own one of the addresses, the transaction could not have been verified without you, which may count as "ownership" if you want to call it that way.

Proving that you own an address can be done by signing a message using the private key of that address. Most wallets provide signing functionality. Just agree on a message, something like "I, [name with which the other party knows you], am the owner of address [address from the input]." After signing, you can send the message and the signature to the other party and he will be able to verify your ownership of the private key.