the public acknowledgment of what God has done: openly declaring his rescue before others, in word and in the offering that goes with it
This word names a single act seen from several angles. At its center is a person who was in trouble, was rescued by God, and now stands up before the community to say so out loud. The same word covers the spoken declaration, the song that carries it, the sacrifice offered at the temple to mark the rescue, and even the choir that performs the whole thing. English Bibles usually shrink all of this to "thanksgiving," but the word is less about a private feeling of gratitude and more about going public: making known, before witnesses, the specific thing God did.
That is why the same word can sit in a sacrifice law and in a worship song without changing its meaning. In the law of offerings it is the rescued person's sacrifice, eaten the same day at a shared meal so that a crowd has to be invited and the rescue gets told. In the worship songs it is the testimony itself, the voice raised to report God's deed. In one striking case it is the name for two great processional choirs marching on the city wall. These are not separate definitions but one act wearing different clothes: the meal, the song, the choir, and the spoken word are all ways of acknowledging God in the open.
A related use turns the same idea toward owning the truth about oneself. When a guilty person is told to acknowledge God by confessing what he has done, the word reaches for that honest admission as well. Whether the truth being acknowledged is God's saving deed or one's own failure, the act is the same: giving God his due by saying out loud what is really so.