or should the two parties be working to negotiate on the terms of the agencies original contract?
The agency's original contract is merely an offer. You can reject their offer and present a counter offer in any format you wish. Most likely, if they have room for negotiation, the agency will be more receptive to minor edits on their original contract. Generally speaking, contract negotiations go better when the series of counter offers involve the minimum changes from the most recent necessary to effect the desired resulting changes. But, in negotiations, the only rule is that there are no rules. So you can toss out their entire agreement and start over with your own if you want.
If you only want to make minor edits, a good technique is to simply mark a line through the language you want to delete, write in the language you want to add, then initial it before signing. If the other side also initials and signs, then the language you edited will be your agreement.
For more extensive changes, you could add a sentence in the original contract that says you will also agree to an addendum to the contract. This could be a set of terms you wish to add to the original agreement to protect yourself. You could and would also want to add a sentence in the original agreement that where the addendum conflicts with any other provision of the agreement, the addendum will control.