to reign as king: to hold the throne and exercise royal power over others
This is the verb behind the noun for a king. Where the noun names the person who wears the crown, this word names what that person does: to sit on the throne and rule. It covers both taking the throne and holding it, so a phrase like "began to reign" and a phrase like "reigns forever" are both this same word.
In the New Testament its subjects range widely. Ordinary human rulers reign, as Archelaus did in Judea. God and Christ reign, and their rule has no end. Strikingly, the redeemed are also said to reign, sharing in Christ's royal rule now and in the age to come.
Paul then stretches the word in a bold picture. In Romans he speaks of death, sin, and grace as if they were kings on a throne, ruling over human life. This is not a different meaning but a vivid image: these forces dominate a person the way a king dominates a realm, and the gospel is the story of one reign replacing another.
English "reign" fits the word closely in every case, which is why it rarely needs explaining. The one thing worth noticing is the active force of it: to reign here is to exercise real power, not merely to hold a ceremonial title.