The other day I wanted to send someone proof that a transaction has gone through. A screenshot would have been the obvious choice, but of course, my banking app wouldn't let me do it.

A screenshot would also be trivial to counterfeit. That being said, I am not aware of any banks that provide any actually tamper-proof, shareable transaction confirmations.

Why does a third party want to know the transaction occured?

These seems a bit like a scam. Why can't they ask the recevier?