(I realize I’m not really replying to you but to whomever you’re quoting and paraphrasing this month. Thread readers should take this as replying about the principles expressed, therefore, not replying to Cole per se.)
Not only are the relevant words never used that way in the New Testament (we or our sins are propitiated, not God – though admittedly that’s entirely opposite to the natural religious expectation that the deity is who has to be propitiated), but the concept that God the Son has to propitiate the wrath of God the Son is nonsense; and the concept that God the Father has wrath against us, so has to be propitiated by God the Son is even more nonsensical. No trinitarian theist should ever take such positions.
(Edited to add: that God has wrath against us, sure, fine, not a problem for trinitarianism. That God opposes God regarding God’s wrath for us, very much not trinitarian theism. Or even non-trinitarian Christianity of any kind that I can think of offhand. Nor is the idea that God is intrinsically wrath against sin an idea coherent with trinitarian theism; nor is the idea that God does love but isn’t essentially love and therefore can choose not to love or to stop loving rational creatures – that’s nothing other than a tacit repudiation of ortho-trin. Pitting God’s love and justice against each other in God so that God does justice to one set of creatures and God does love to another set of creatures, also super-bad trinitarian theology. I would and did reject such positions, and several others like it, explicitly as a coherent orthodox trinitarian theist, before I became a Christian universalist. From which rejections and corrections, I became a Christian universalist – but as a corollary to ortho-trin. That level of theology has to come first, soteriology follows.)
It also doesn’t fit Calv notions of election, since God the Father already loves the elect (and only the elect) so much to send the Son to die for us while we are yet sinners; which is exactly opposite to the idea that God’s wrath has to be “propitiated” for the elect. Whereas on Calv notions of the non-elect, God doesn’t have (at least) saving love (and maybe no real love at all, Calv theologians disputing this point with each other) for the non-elect, so God’s wrath was never propitiated for them at all, nor did any Person of God even intend to propitiate the wrath of another Person of God for them (assuming such a schism of intention was even possible within ortho-trin theism, which it is not).
Except God the Father (and God the Holy Spirit for that matter) originally participates with giving grace to sinners and cleansing our sins, so at best we now have God the Father pleasing God the Father in order to turn away the wrath of God the Father. Even on a non-trinitarian theism this would be schizophrenic.
The mercy-seat concept of the term ‘propitiation’ (and its Greek original) is accurate; but all the Persons sit upon the mercy seat, and are already smiling (per the Greek term) and leaning forward (per the Latin term) to clean sinners who agree to cooperate with the cleaning – a cooperation only possible if God reaches out first to provide them with the capability to even agree to cooperate with the cleaning. On Calv theology God never reaches out like that for the chosen non-elect at all anyway (by God’s authoritative choice); but in regard to those elected for salvation (on any nominally trinitarian theology, Calv, Arm, or Kath) God doesn’t have to propitiate God’s wrath before or after reaching out from the mercy-seat of judgment to have mercy on sinners.
Except that when the term translated “imputation” occurs in the scriptures, it refers to an accurate accounting of the situation, not to a useful legal fiction which would be analogically tantamount to cooking the accounting books or embezzling. God of all people doesn’t pretend God is something God is not.
Fortunately, the scriptures don’t say that God (in any Person of God) imputes our sins to God (including in 2 Cor 5:21). Though on Calv theology our sins are in fact imputed to God (except when that corollary becomes theologically inconvenient), so I guess the theological inconsistency there is consistent in its own weird way. 
There is some scriptural evidence that we as sinners impute our sins to God, so again in a backhanded way this version of PSA fits scriptural testimony – but not in any way which says the standard PSA proponents are right to be doing so!
Calvs (and Arms to be fair) like to say that on a regular basis as though the people they’re talking to don’t believe or remember this. But that doesn’t make the arguments being salted with this saying any more coherent or accurate.
Mere punishment for legal purposes is meaningless, and puts God at the mercy of God’s own law at best: well, sin is really really serious, so even though God forgives the sinner and agrees not to punish him, God just has to punish someone in order to be just, so God punishes Himself – He doesn’t sweep these serious sins against Him under the rug because not to punish someone, even if the someone is completely innocent, would be unjust (the punishment of the innocent being justice instead of injustice on this account), and the demeaning of God would be endorsed if He didn’t find someone innocent to punish instead of the guilty! But since He finds an innocent person to punish instead of the guilty, now a lie will not reign (on this theory) at the core of reality.
Except for the lie reigning at the core of reality on this theory.
It would look a little better if the notion were more coherently kept in mind that God is punishing Himself for sins done against God – at least it would be that much more trinitarian – but that wouldn’t change the mere legalism involved. Nor would it comport very well with the intrinsic justice of God between the Persons from all eternity as the ground of all reality: the essential reality of the justice of God does not involve the Father pretending the Son is unjust and punishing Him as though the Son is unjust (much less does it involve the Son actually being unjust and being punished by the Father).
Beyond all this, on even non-trinitarian Christologies, much less on any coherent trinitarian Christology, the Father doesn’t punish the Son with some kind of hopeless punishment, whether eternal conscious torment or annihilation. So even on this merely legalistic account, the Son isn’t punished with the just punishment required by God’s justice, but with a lesser punishment! – God’s “just” requirements remain unmet! – the (merely) “legal demands of the law” remain un-canceled!
Well, then, if someone has raped you, you should pretend you raped yourself and put yourself into eternal conscious torment in a prison somewhere, in punishment of yourself, so that you can forgive the person who raped you; since without your hopeless (or hopeful?) punishment of yourself your wrath against your rapist will never be propitiated and you’ll never be able to forgive him.
On the Calv account you’re (going back to) accepting now, that’s what God does for those whom He intends to forgive; whereas the only people who actually go to prison are those He never even intended to forgive in the first place (by His own authoritative choice. Also they were incapable of doing anything other than sinning against Him, by God’s authoritative choice to bring them into existence without those capabilities, and to sustain them in existence without those capabilities which He could have chosen to give them but chooses not to do so.)
The Bible does teach this; it does not teach that the sins of sinners are accurately reckoned to Christ (or even that God pretends that they are accurately reckoned to Christ). In fact, on the standard model of PSA, there is less than no reason at all for anyone to be crucified with Christ, to die with Christ so that we may rise with Christ: Christ was supposed to be (merely) substituted for some (not all, per Calvs) sinners.
Except that even on this (highly broken) soteriology, everyone was outside the gate and didn’t want the true God but only wanted to stay evil forever; so If the only proper justice for such people is for such people to be punished instead of the most innocent person being punished instead, then every sinner should be punished forever so that the brightness of God’s infinite glory shines forever. Even the merciful salvation of Christ becomes improperly unjust on this theory – and moreso again because Christ the innocent is not hopelessly punished forever as an evildoer!
The end result of such a purely schizophrenic plan is that the brightness of God’s infinite glory is permanently and infinitely defeated by God’s own authoritative choice!
(And then the supposed “glories” of “this justice” supposedly makes “His love shine all the brighter.”)
So, the other part being schizophrenic nonsense (God’s main purpose in punishing people being to ensure that His infinite glory fails in several ways, which by the magic of wishful thinking also counts as His infinite glory shining forever), someone goes on to say this sort of thing instead. Now God’s main purpose, since the other main purpose clearly and in several ways fails by God’s own authoritative choice, is to protect His actual children instead of His children who aren’t His children (by nothing other or greater than God’s own authoritative choice). God will “do anything” to protect the children He chooses to protect, except to actually protect them from such monstrous evil, since such monstrous evil only exists by God’s authoritative choice in the first place (unless we’re hewing straight into God/anti-God dualism, which isn’t even supernaturalistic theism, much less trinitarian theism); and since the children He chooses to save from sin are still afflicted by monstrous sin to start with, and they continue being afflicted by it, at least externally and (per even St. Paul also internally, though there is dispute about this among theologians) until God sequesters off the other children He chooses never to save from sin (or possibly annihilates them out of existence). Leaving the other children behind who shall be forever and only ever afflicted with monstrous sin: by God’s authoritative choice.
That people are taught to call, and to even value, this idea as beautiful indeed… is not beautiful, indeed.
The people you’re getting all this from, Cole, are spazzing theologically all over the place, and are doing you no favors thereby. You yourself used to realize this, or so you said. I’m sorry they’ve got hold of you again. But the truth, and the infinite glory of God’s actual eternal and positively righteous justice, will come forth eventually. And that will be beautiful indeed. 
Jason,how can our sins be appeased? Isn’t that the meaning of “propitiate”? Or who would be appeasing or placating us?