Written contract

what you mean is a “contract memorialised in writing”. A contract is a contract it 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, 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.

See also

• Scribam, ergo salvum sum