burning anger: hot, intense wrath that boils up and demands an outlet
This is by far the most common use of the word. The root behind it means to be hot, so the picture under the wrath is literal heat: anger that has come to a boil and is felt as a fire inside. It is not cool displeasure but a fierce, pressing rage, the kind that flushes the face and wants to act.
The word is used both for God and for people, and the same heat is meant in each. When it is God's, it names the settled, righteous fury he pours out against sin, often paired with judgment and the language of being poured out or kindled. When it is a person's, it names hot temper, jealousy, and the rage that flares in a quarrel.
The two readings below are one meaning seen from two sides, not two different ideas. The shading that matters is simply whose anger it is, because God's burning anger is just and measured while a person's is usually the very thing the wisdom books warn against.