to exult: to break into open, out-loud joy over a rescue, a victory, or the righteous coming out on top
This is a rare, poetic verb. It appears only eight times, always in a song, a psalm, or a proverb, and never in a plain report of what somebody did. It names joy at full volume, the kind that has to come out, and it almost always rises at a particular moment: something has just gone right. A rescue has landed, a threat has passed, or the good side has finally come out on top. The everyday word for being glad (שָׂמַח) covers happiness of every kind; this one is the shout that goes up when the score is settled.
The word keeps close company with people who are in the right. Three of its eight uses put the righteous directly in the line: the upright celebrating in God's presence, a whole town celebrating because good people are prospering, and the righteous themselves in their moment of triumph. That is why English versions reach for "triumph" here about as often as they reach for "rejoice". This is not free-floating good cheer but gladness with a reason behind it, and the reason is usually that things have turned out the way they should have.
Where the joy is aimed decides its whole character. Turned toward God it is worship: Hannah's heart exults in the LORD, everyone who loves God's name is invited to do the same, and even the open countryside is called to join in when God comes to set the world right. Turned toward a person who has been brought low it becomes gloating, which is why one psalm prays that enemies will never get the chance. Hebrew has three sound-alike verbs in this territory, this one along with עָלַז and עָלַס, and a poet may be choosing among them partly for how the line sounds, so do not press a hard distinction where the difference may just be music.