I found the article ambiguous on what it means by covenant and covenant of works.
I would argue that all three covenants with God (Adam/Eve, Israel, us) depend for their existence and continuation purely upon grace, but that they commit us to obedience and that there are consequences within the covenant for sin. I would call them (or it -- since I prefer the traditional approach of seeing one covenant in different forms) all a covenant of grace, because its existence depends upon grace, and God treats us with grace within the covenant. However they certainly commit us to obedience, and violation of that commitment has, within the provisions of the covenant, disciplinary consequences.
In particular, I don't see that Adam and Eve were under a different kind of covenant from anyone else. God loved them based on the fact that they were his creation in his image. There's no sign that they had to earn that. But disobedience had consequences, just as it did for Israel and does for Christians. In the case of Eden it was ejection from Eden (and possibly mortality, though Calvin speculates that if they had been obedient they would still have ended their lives on earth, just not in the same way).
Similarly today, the new covenant exists by grace, and God treats us with grace, but there is still accountability and discipline, though the form which that takes is different.