Covenant is used in a particular way in scripture. It is used of God’s work of establishing a relationship with fallen man. The one possible reference to a Covenant with Adam is difficult to prove. And considering the use of covenant in scripture, it is difficult to say that Adam was in covenant with God, except by analogy to other covenants. So we argue that this is an implied covenant.
How is covenant used? I would begin with an analogy. Marriage is a covenant. While a mere relationship is not a covenant. So covenant involves a formalized relationship, that is, a relationship that did not exist, but through promises set out with obligations given, something new is formed that was not their before. That is not the relationship with God and Adam, where Adam’s relationship with God is established simply in being made by God and functions more as an analogy between father and son, rather than husband and wife. Later covenants are made in order to establish a relationship with God’s people (Abraham) or are made to restore God’s relationship with his people (God restores covenant with his people after the exile).
So why we can’t we simply extend the word covenant by analogy to other relationships? The problem is that the scriptures already use covenant in a certain way. We either then empty covenant of its content and context until it just means “relationship,” especially when it comes to the next step, an inter-Trinitarian covenant. In this case, succession of covenant is flattened. Or, the scriptural definition of covenant begins to leak into other administrations. The Adamic administration becomes a covenant of works by which Adam must merit eternal life. Or, a covenant between the persons of the Trinity, where the language moves more and more toward a social trinity, even a tri-theism.