This summary is not the rule. The rule text controls.
If you want the court to award attorney fees because a written contract says the losing side has to pay them, you need to show the court that contract. You must give the court the original document or a true copy of it when you submit your proposed judgment. If a true copy was already attached to your court papers or written out in them, you don't need to submit it again.
This same requirement applies if you're claiming fees under ORS 20.096, which lets a party claim attorney fees even when the contract only mentions fees for one side. The idea behind this law is that if a contract allows fees for one party, the other party can claim them too under certain conditions.
If you don't have the original document or a copy, you're not out of luck. The court can accept other proof instead, like a sworn statement (affidavit) or testimony explaining what the contract says and why you can't produce it. This gives people a way to support their fee request even when the paperwork isn't available.