The most useful fact about the atonement is a negative one, and it is almost never stated: no ecumenical council has ever defined a theory of it, and no creed specifies a mechanism.
This is genuinely odd. The same councils that left the atonement alone defined the person of Christ to a degree of precision that split the church permanently. Chalcedon in 451 produced four adverbs, without confusion, without change, without division, without separation, to rule out four specific proposals about how divinity and humanity are united. That is a body prepared to legislate on metaphysics in considerable detail.
On the question of what the death of Christ accomplishes and by what means, the same tradition says only that he suffered "for us and for our salvation" and stops.
What follows from the silence
Three things, and they are worth keeping distinct.
No theory of atonement is heresy. A Christian may hold ransom, satisfaction, moral influence, penal substitution, recapitulation or Christus Victor without departing from any conciliar definition. Denunciations across this line are denunciations by one confession of another, not applications of a settled rule.
The theories are not all answering the same question. Ransom and Christus Victor ask what the death defeats. Satisfaction and penal substitution ask what it discharges. Moral influence asks what it shows. Recapitulation asks what it heals. Much of the argument between their advocates consists of treating a rival as a poor answer to a question it was not asked.
Several are routinely held together. The same preacher will describe a victory over death at Easter and a debt discharged on Good Friday without noticing a change of model, because the tradition never required a choice.
Why it is contested now rather than historically
The live dispute is about penal substitution, and it is recent. The theory is a Reformation development and became the working assumption of Anglophone evangelicalism. From the later twentieth century it has drawn sustained criticism, that it sets the Father against the Son, that it makes pardon conditional on someone being punished, and that it has been used to give innocent suffering a redemptive gloss. Its defenders answer that the criticism misreads a trinitarian act as a transaction between two parties.
That argument is in progress. An entry that resolved it would be reporting a result that does not exist.