The nature of 'He paid our debt'

One of the biggest problems the modern Calvinist has is a dreadful ignorance of the way Christ paid our debt.  I think this is in large part thanks to John Owen who encouraged his readers to adopt a monitary approach.  As an example of this error, see here.
I can't think of any better way to correct this very common deficency in the modern Calvinist thinking than by quoting Dabney46, from chapter 35 of his systematic theology.

In a mere pecuniary debt, the claim is on the money owed, not on the person owing. The amount is numerically estimated. Hence, the surety, in making vicarious payment, must pay the exact number of coins due. And when he has done that, he has, ipso facto, satisfied the debt. His offer of such payment in full is a legal tender which leaves the creditor no discretion of assent or refusal. If he refuses, his claim is canceled for once and all.

Dabney goes on to give an example.

[Say] a mechanic is justly indebted to a land owner in the sum of one hundred pounds and has no money wherewith to pay. Now, should a rich brother offer the landlord the full hundred pounds, in coin of the realm, this would be a legal tender. It would, ipso facto cancel the debt, even though the creditor captiously rejected it. Christ’s satisfaction is not ipso facto in this commercial sense.

The error is obvious, if Christ paid the debt from the cross then our obligation was discharged at the cross.  Which means we have no need for faith, and as a bonus, all those warnings Jesus gave about the need for faith are wrong too.  Dabney continues,

But the legal claim on us for obedience and penalty is personal....Christ’s satisfaction cannot be forced on the divine Creditor as a legal tender; it does not free us ipso facto. And God, the Creditor, has an optional discretion to decline the proffer, if He chooses (before He is bound by His own covenant), or to accept it. Hence, the extent to which, and the terms on which Christ’s vicarious actions shall actually satisfy the law, depend simply on the stipulations made between Father and Son, in the covenant of redemption.

That is to say, we must have faith. Faith is the channel by which Christs righteousness flows into our lives, it's the open hand- that is the requirement.  God demands from everyone faith, because we are naturally capable of it even after our fall.  Lastly, I'll quote Dabney's example of the moral payment:

There is a second supposition, the kind brother is not rich, but is himself an able mechanic, and seeing that the landlord is engaged in building, he proposes that he will work as a builder for him two hundred days, at ten shillings per day, to cancel his poor brother’s debt. This proposal, on the one hand, is not a "legal tender," and does not compel the creditor. He may say that he has already enough mechanics, who are paid in advance, so that he cannot take the proposal. But, if he judges it convenient to accept it, although he does not get the coin, he gets an actual equivalent for his claim, and a fair one... The debtor may thus get a valid release on the terms freely covenanted between the surety and creditor. 

 This is not to deny that we owe God a debt, and God has a right to collect on it, somewhat like money owed, but it is to deny that Christ pays out debt in this way.