You are thinking of a social agreement often called a “handshake agreement” or “verbal agreement”. Yes a verbal or social agreement at law is often superceded by a written agreement mainly because there’s a specific common clause in written agreements that say exactly this.

The above legal concept is distinct from the term “social contract” that is a sociological term and has a wider definition that a mere legal meaning. Most of our laws are intended to capture concepts that have wider social and cultural relevance to the society to which they apply.

“Marriage” by example is a term with both a distinct legal meaning in many legal jurisdictions as well as a wider sociological conception that can vary from culture to culture.