Owen:

נָשׂא (nasa) also, verse 12, arguing a taking of the punishment of sin from us and translating it to himself, signifies as much, yea all that we do by the word satisfaction. So also doth that of ἀνήνεγκεν, used by Peter in the room thereof: for to bear iniquity, in the Scripture language, is to undergo the punishment due to it, Lev. v. 1; which we call to make satisfaction for it;—which is farther illustrated by a declaration how he bare our sins, even by being “wounded for our transgressions, and bruised for our iniquities,” Isa. liii. 5; whereunto is added, in the close, that “the chastisement of our peace was upon him.” Every chastisement is either νουθετική, for instruction, or παραδειγματική, for example, punishment and correction. The first can have no place in our Saviour; the Son of God had no need to be taught with such thorns and briers. It must, therefore, be for punishment and correction, and that for our sins then upon him; whereby our peace or freedom
from punishment was procured.

John Owen, “The Death of Death” in, Works, 10:266; or, Death of Death, 154. [Some spelling modernized and underlining mine.]

Hodge:

1) 1. To impute is to reckon to, or to lay to one’s account. So far as the meaning of the word is concerned, it makes no difference whether the thing imputed be sin or righteousness; whether it is our own personally, or the sin or righteousness of another.

2. To impute sin, in Scriptural and theological language, is to impute the guilt of sin. And by guilt is meant not criminality or moral ill-desert, or demerit, much less moral pollution, but the judicial obligation to satisfy justice. Hence the evil consequent on the imputation is not an arbitrary infliction; not merely a misfortune or calamity; not a chastisement in the proper sense of that word, but a punishment, i.e., an evil inflicted in execution of the penalty of law and for the satisfaction of justice.

3. A third remark in elucidation of what is meant by the imputation of Adam’s sin is, that by all theologians, Reformed and Lutheran, it is admitted, that in the imputation of Adam’s sin to us, of our sins to Christ, and of Christ’s righteousness to believers, the nature of imputation is the same, so that the one case illustrates the others. When it is said that our sins were imputed to Christ, or that He bore our sins, it is not meant that he actually committed our sins, or that He was morally criminal on account of them, or that the demerit of them rested upon Him. All that is meant is that He assumed, in the language of the older theologians, "our law-place." He undertook to answer the demands of justice for the sins of men, or, as it is expressed by the Apostle, to be made a curse for them. In like manner, when it is said that the righteousness of Christ is imputed to believers, it does not mean that they wrought out that righteousness, that they were the agents of the acts of Christ in obeying the law; nor that the merit of his righteousness is their personal merit; nor that it constitutes their moral character; it simply means that his righteousness, having been wrought out by Christ for the benefit of his people, in their name, by Him as their representative, it is laid to their account, so that God can be just in justifying the ungodly. Much of the difficulty on this subject arises from the ambiguity of language. The words righteous and unrighteous have two distinct meanings. Sometimes they express moral character. A righteous man is an upright or good man. At other times, these words do not express moral character, but simply relation to justice. In this sense a righteous man is one with regard to whom the demands of justice are satisfied. He may be personally unrighteous (or ungodly) and legally righteous. If this were not so, no sinner could be saved. There is not a believer on earth who does not feel and acknowledge himself to be personally unrighteous, ill-deserving, meriting the wrath and curse of God. Nevertheless he rejoices in the assurance that the infinitely meritorious righteousness of Christ, his full atonement for all sin, constitutes Him legally, not morally, righteous in the sight of divine justice. When, therefore, God pronounces the unrighteous to be righteous, He does not declare them to be what they are not. He simply declares that their debt to justice has been paid by another. And when it is said that the sin of Adam is imputed to his posterity, it is not meant that they committed his sin, or were the agents of his act, nor is it meant that they are morally criminal for his transgression; that it is for them the ground of remorse and self-reproach; but simply that in virtue of the union between him and his descendants, his sin is the judicial ground of the condemnation of his race, precisely as the righteousness of Christ is the judicial ground of the justification of his people. So much for the statement of the question. Charles Hodge, Systematic Theology (Wm. B. Eerdmans: Michigan: 1981), 2:194-195.

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Baxter:

1) Whether Posterity be Guilty of Death, by Reason of the Actual Sins of their Immediate Parents?

As little as is said by divines on this question, it is no over-curious or needless unprofitable subject, but weighty and needful to be understood by all Christians that can reach to the understanding of it. For as it is useful for the opening cause and nature of original guilt, so, if it should prove true that we are guilty by the sins of our immediate parents, it would be necessary that we know it, for our due humiliation, and that we may in penitent confessions and deprecations prevail with God for the pardon thereof. As it is thought a dangerous thing to deny original sin, because they that so do, will not be humbled under it, and sensible of their misery by it, nor of the necessity of God’s mercy, or Christ’s blood for the pardon of it, nor will apply themselves to God by Christ in faith, confession, and prayer for pardon, and consequently are in danger of missing of pardon, so in the present case, the same reasons will prove it as well dangerous to deny our guilt of our parents’ sins, if, indeed, we are so guilty. Which that we may inquire into, after a very brief explication of the terms of the questions, I shall lay down a few necessary distinctions, and then assert what I judge to be the truth in certain propositions, and prove such of them as most require proof.

1. By [immediate parents], we mean those that personally beget: by [posterity] we mean their children begotten. By reason of [actual sin], we mean, by the merit of those sins which our parents themselves committed, or by a resultancy from such sin compared with the rule. By guilt, we mean obligation to punishment. By death, we mean the destruction, or final misery of the creature, either death temporal or eternal.

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Richards:

Second. Was his death, then, of vicarious import simply? or was it strictly vicarious?

That it was of vicarious import cannot reasonably be denied, if we compare it with the legal sacrifices, or attend to the express language of Scripture on the subject.

The victims under the law were vicarious offerings; they suffered in the room and stead of the offerer, and thus far there was a transfer, not of sin or guilt, strictly speaking, but of its penal effects; suffering and death, only, were transferred, and this is what is meant by putting the iniquities of the sinner upon the head of the victim, and of the victim’s bearing the iniquities of the sinner.

To suppose a literal transfer, either of sin or of punishment, would be to suppose something which is entirely unauthorized by the language of Scripture, and at the same time to involve the absurdity of making a man and even a beast guilty by proxy. Sin, guilt, ill-desert, are in the very nature of things personal; and punishment presupposes guilt, and guilt in the subject; neither the one nor the other is properly transferable. Or, to use the language of Magee: “Guilt and punishment cannot be conceived but with reference to consciousness which cannot be transferred.”

While we would maintain, therefore, that the sufferings of Christ were of vicarious import, because he suffered in the room of sinners, and bore the indications of Divine wrath for their sakes, we cannot subscribe to the opinion that they were strictly vicarious, if by this is meant that the sins of those for whom he suffered, their personal desert and their punishment were literally transferred to him. We maintain the doctrine of substitution, but not such a substitution as implies a transfer of character, and consequently of desert and punishment. This we think to be impossible; and unnecessary, if not impossible. It was enough that there should be a transfer of sufferings, and these, not exactly in kind, degree, or duration, but in all their circumstances amounting to a full equivalent in their moral effect upon the government of God. We hold that Jesus died in the room of the guilty, that though innocent himself, he was made sin for us, or treated as a sinner on our account, and in our stead; that the Lord laid on him the iniquities of us all, and that he bore our sins in his own body on the tree, by suffering what was a full equivalent to the punishment due to our offences. But this, we think, is all the substitution which the Scriptures teach, all that the nature of things will admit, and all that was necessary to effect the same moral ends in the government of God which would have been effected by inflicting on the transgressor the penal sanctions of his law.

James Richards, Lectures on Mental Philosophy and Theology (New York: Published by M.W. Dodd, 1846), 312-314.