Written contract
what you mean is a “contract memorialised in writing”. A contract is — in Daniel Dennett’s marvelous term — “substrate-neutral”. The paper on which a contract’s terms are set out is a record of an immaterial thing. The paper is the cave wall; the contract is the Platonic form. This might seem a fatuous distinction — especially where someone has had the sense to provide that the written thing is a comprehensive and final record of the parties’ respective obligations[1] or where the contract may not be varied but by further written agreement — but it can help navigate your way out of odd situations such as those where the parties think they have signed an agreement but haven't, or they did sign the agreement, but have subsequently lost it.
Both happen.