Damage was caused by BOTH of us, if I go and get an estimate to repair damage on my vehicle, even though it is small, it'd be about the same amount if not even more

And that matters because?

The contract was giving money to someone for specific purpose and that person agreeing to what's in it and signing that contract. All I want is to be sure amount I paid is not more than the damage on the vehicle, and his intention is to repair the vehicle, signed/said in front of witnesses, email proof etc. so I just want to make sure that's done

Then you should have required a written estimate before handing over money.

Look you came here asking for advice. You have been given that advice. Yet you want to keep arguing. The law will not look on that paper as a contract. They will view it as a waiver of future liability. There is no two way here. You damaged his car and agreed to pay for the damage. You (rightly) asked for him to sign a waiver to agree to accept your payment as payment in full. You cannot require that he make the repair. Even though it was included in the waiver its unenforceable.

You have no case. End of story!