A contract is a contract is a contract.
A circular argument (not a fallacy). The problem is the definition of 'contract'.
This word first appeared in the English lexicon around the 14th century. It is a melding of the Latin word 'tract' (meaning passage of time), and 'con' (with). The word originally meant an agreement between two people for a period of time. At that time, of course, few things were written down. Contracts were often denoted with a token of the promise made, often a knife, since people in the town knew what the knife meant and how it was obtained. Such tokens were indicative of a title to land, a rental agreement, a production agreement, etc. Many written documents were forgeries back then.
You might say even the rainbow itself is a token of contract, between God and his people. The promise is that the world would never be flooded again.
Contracts today can be verbal. They can be written down. They can be simple or they can be complex. They can be marked by a token of the promise even today. It is quite possible for a mere verbal contract to be enforceable by a court of law. Different States use different tests for this.
Since the 40's, the word 'contract' was also used to denote a promise to kill someone. They were put out by mobs when hiring hit men. These were verbal agreements to commit murder for pay. Crossing them was dangerous, since either party would put the offender on a contract in the same way. Writing them down was pointless and futile, since courts don't enforce criminal activity, but they were contracts nevertheless.
Contracts do not need to be put into legal format (like you see in court documents). They can verbal agreements or written on any tatty piece of paper, etc. The written ones may or may not be notarized.
So:
A contract is simply an agreement, or promise over time. That is it. That is all. It can be verbal, written down, denoted with a token of some kind, anything. At some time an agreement of a promise of action over time is made. That is a contract, even if it later falls apart.