Hoops, for example, if I can sort through the coy ambiguity, seems to offer the following definition of Christian: one whose sins are forgiven. I assume this forgiveness comes through acceptance of the atonement of Christ. (I'm sure Hoops will say I have this all wrong, and am lying to boot.)
No, I would not say you're lying. I would say you misunderstand, though. Yes, a Christian is one whose sins are forgiven. It is by the power of Christ that this occurs. I don't know that acceptance of the atonement in the sense that I think you mean it has much to do with it at all.
It seems to me that if a religion teaches that Jesus Christ's atonement is the only way to reunite human beings to God, and teaches utilizing the power of the atonement to engage in personal repentance, then that is a Christian religion
Of course. I don't know what you mean by "utilizing the power" but in a broad sense, yes. However, if a religion does not teach who Jesus really is, then it is teaching a lie. It is teaching non-Christian.
even by Hoops' definition.
Really?
Mormonism certainly does that. And yet it isn't enough.
It certainly does not. It has taken God and morphed him into something unrecognizable when viewed through Christian orthodoxy.