to be king: to hold and exercise royal rule, or to come to the throne
This is the verb behind the Hebrew word for king. It covers the whole arc of holding a throne: stepping up to it and then ruling from it. In the books of Kings and Chronicles it is the workhorse of the royal records, the verb in the recurring summary that a man became king over Israel or Judah and reigned so many years.
The two shades below are not different words but two moments of one royal life. One looks at the start of a reign, the moment a person takes the throne. The other looks at the reign itself, the ongoing exercise of rule. Hebrew often leaves the surrounding sentence to tell you which is in view, and many verses sit comfortably with either.
The most important use is for God. When Scripture says the Lord reigns, it uses this same ordinary royal verb, claiming for God the very thing kings claim: real, active rule. That is why the song at the sea ends, the Lord will reign forever and ever.