right: belonging to the right-hand side, and so the side of honor and power
At its plainest, this word simply means the right side as opposed to the left: the right eye, the right hand, the right cheek, the right ear, the right side of a boat. In a world where almost everyone was right-handed, the right side was the strong side, the working side, the side you reached out with.
From that everyday strength the word picks up a settled social meaning. To stand or be seated at someone's right was the place of greatest honor next to that person, and the seat from which their authority was shared. The New Testament leans on this constantly to describe Jesus after the resurrection, seated at the right hand of God: the single most honored place in the universe, where he reigns and intercedes. The same picture lies behind the request of James and John for the seats on Jesus' right and left, and behind the sheep set on the king's right in the judgment scene.
The right hand also carried the weight of a promise. To extend it and clasp another's was to pledge agreement and partnership, which is why Paul speaks of being given the right hand of fellowship.
These are not different words but one word stretched along a single line: the strong side becomes the side of honor becomes the hand of commitment. The reader rarely has to choose between them, but knowing the social charge of the right side keeps the honor passages from sounding like mere geography.