In Reformed theology, what exactly does the doctrine of "limited atonement" actually assert?

"Is it the case that a substantial proportion of those theologians and church leaders today who consider themselves of a traditional Reformed persuasion would be willing to affirm the following five statements about the atonement?"

Today's Reformed world is, as you have noted in your comments re: Piper, a bit broad. In general, your five points accurately reflect the traditional, limited atonement view of Calvinism in particular. (See Institutes, Book III). (Arminians are also Reformed and differ considerably from Calvinists in their soteriology). It remains the belief of a substantial proportion of Calvinist leaders - Christ died only for the Elect and there was never any intention of Him dying for anyone else. But, as you have noted, there are some so-called Four-Pointers who teach that the atonement was not limited, but does, in fact, extend to all persons even though many of those are not Elect and will not be saved. Pre-Victorian Calvinists would have agreed with your five points.

References: 1. https://www.crossway.org/articles/10-things-you-should-know-about-definite-atonement/ 2. Calvin, Institutes 3. 1689 Baptist Confession which reads in part: Paragraph 6. As God hath appointed the elect unto glory, so He hath, by the eternal and most free purpose of His will, foreordained all the means thereunto;13 wherefore they who are elected, being fallen in Adam, are redeemed by Christ,14 are effectually called unto faith in Christ, by His Spirit working in due season, are justified, adopted, sanctified,15 and kept by His power through faith unto salvation;16 neither are any other redeemed by Christ, or effectually called, justified, adopted, sanctified, and saved, but the elect only.

For any of the so-called five points of Calvinism, it is usually best and clearest to go to their original expression in the Canons of Dort. In the 17th century, the followers of Arminius promulgated five points of doctrine on which they dissented from the reformed position. In response, reformed delegates gathered at Dort and drew up responses to each of the five points. These are what are now termed the "five points of Calvinism." They were arguably taught by Calvin, although not as explicitly as at Dort, and they were present in the pre-existing reformed confessions (the Belgic Confession and the Heidelberg Catechism), and they are present in the Westminster Standards (containing the Westminster Confession of Faith, the Westminster Larger Catechism, and the Westminster Shorter Catechism, which post-date the aforementioned confessions but are considered to teach the same doctrine as the Three Forms of Unity).

So the Canons of Dort have five heads of doctrine, corresponding to the five points of Remonstrance that they were convened to respond to. The second head defends what is often called "limited atonement."

To your question, "How many of the following five statements would a typical self-identifying adherent of Particular Redemption be willing to affirm?" the short answer is that adherents of the canons of Dort would affirm all five statements.

Taken in order:

Particularity in motivation (the first statement): that God's will for Christ to die was not a love toward humanity as a whole, nor for each individual sinner, but for the elect only.

This can be seen in article 8 of the second head.

For this was the sovereign counsel, and most gracious will and purpose of God the Father, that the quickening and saving efficacy of the most precious death of his Son should extend to all the elect, for bestowing upon them alone the gift of justifying faith, thereby to bring them infallibly to salvation: that is, it was the will of God, that Christ by the blood of the cross, whereby he confirmed the new covenant, should effectually redeem out of every people, tribe, nation, and language, all those, and those only, who were from eternity chosen to salvation, and given to him by the Father; that he should confer upon them faith, which together with all the other saving gifts of the Holy Spirit, he purchased for them by his death; should purge them from all sin, both original and actual, whether committed before or after believing; and having faithfully preserved them even to the end, should at last bring them free from every spot and blemish to the enjoyment of glory in his own presence forever.

Particularity in purpose (the second statement): that all of the potentiality of the atonement's benefits inevitably find actuality, so that none of those Christ died for will be lost.

This can be seen in articles 5, 8, and 9 of the second head:

The promise of the gospel is, that whosoever believeth in Christ crucified, shall not perish, but have everlasting life. ... For this was the sovereign counsel, and most gracious will and purpose of God the Father, that the quickening and saving efficacy of the most precious death of his Son should extend to all the elect, for bestowing upon them alone the gift of justifying faith, thereby to bring them infallibly to salvation. ... This purpose proceeding from everlasting love towards the elect, has from the beginning of the world to this day been powerfully accomplished, and will henceforward still continue to be accomplished, notwithstanding all the ineffectual opposition of the gates of hell, so that the elect in due time may be gathered together into one, and that there never may be wanting a church composed of believers, the foundation of which is laid in the blood of Christ, which may steadfastly love, and faithfully serve him as their Savior, who as a bridegroom for his bride, laid down his life for them upon the cross, and which may celebrate his praises here and through all eternity.

But it is most clearly seen in the third and fourth heads (which are combined into one), articles 10 and 14:

But that others who are called by the gospel, obey the call, and are converted, is not to be ascribed to the proper exercise of free will, whereby one distinguishes himself above others, equally furnished with grace sufficient for faith and conversions, as the proud heresy of Pelagius maintains; but it must be wholly ascribed to God, who as he has chosen his own from eternity in Christ, so he confers upon them faith and repentance, rescues them from the power of darkness, and translates them into the kingdom of his own Son, that they may show forth the praises of him, who hath called them out of darkness into his marvelous light; and may glory not in themselves, but in the Lord according to the testimony of the apostles in various places. ... Faith is therefore to be considered as the gift of God, not on account of its being offered by God to man, to be accepted or rejected at his pleasure; but because it is in reality conferred, breathed, and infused into him; or even because God bestows the power or ability to believe, and then expects that man should by the exercise of his own free will, consent to the terms of that salvation, and actually believe in Christ; but because he who works in man both to will and to do, and indeed all things in all, produces both the will to believe, and the act of believing also.

Particularity in the imputation of sin to Christ (the third statement): that the wrath poured out on Christ was wrath against the sins of the elect, not the sins of the reprobate.

This one is harder to find directly stated, but it follows from the others. It's easier to find statements to this effect in the Westminster Confession of Faith. See chapter 11 article 4:

Christ, by his obedience and death, did fully discharge the debt of all those that are thus justified, and did make a proper, real, and full satisfaction to his Father’s justice in their behalf.

In conjunction with chapter 3 articles 6 and 7:

As God hath appointed the elect unto glory, so hath he, by the eternal and most free purpose of his will, fore-ordained all the means thereunto. Wherefore they who are elected, being fallen in Adam, are redeemed by Christ, are effectually called unto faith in Christ by his Spirit working in due season; are justified, adopted, sanctified, and kept by his power through faith unto salvation. Neither are any other redeemed by Christ, effectually called, justified, adopted, sanctified, and saved, but the elect only. The rest of mankind God was pleased, according to the unsearchable counsel of his own will, whereby he extendeth or withholdeth mercy as he pleaseth, for the glory of his sovereign power over his creatures, to pass by, and to ordain them to dishonor and wrath for their sin, to the praise of his glorious justice.

The point in the above sections of the Westminster Confession is that God has ordained all the means to save the elect, starting with the discharge of their debt on the cross. The reprobate have no such means afforded to them.

I'll deal with the last two statements together, since the fourth is in some ways a restatement of the second, and as far as I can tell, the main difference between it and the second statement is that it sets up the fifth. The particularity of the atonement does not invalidate the free offer of the gospel (the fourth and fifth statements).

This is stated quite straightforwardly in the third and fourth heads of doctrine in the Canons of Dort, articles 8 and 9:

As many as are called by the gospel, are unfeignedly called. For God hath most earnestly and truly shown in his Word, what is pleasing to him, namely, that those who are called should come to him. He, moreover, seriously promises eternal life, and rest, to as many as shall come to him, and believe on him. It is not the fault of the gospel, nor of Christ, offered therein, nor of God, who calls men by the gospel, and confers upon them various gifts, that those who are called by the ministry of the word, refuse to come, and be converted: the fault lies in themselves; some of whom when called, regardless of their danger, reject the word of life; others, though they receive it, suffer it not to make a lasting impression on their heart; therefore, their joy, arising only from a temporary faith, soon vanishes, and they fall away; while others choke the seed of the word by perplexing cares, and the pleasures of this world, and produce no fruit. - This our Savior teaches in the parable of the sower. Matthew 13.

I have presented what I take to be the majority view of those who subscribe fully to the Three Forms of Unity or Westminster Standards, but there may be people who would claim to fully subscribe to one or the other and agree with your alternate formulation, that Christ's death legally purchased redemption for all people without exception and yet God has decreed to pass them over. If I'm understanding the view you formulated and the view of a minority of Calvinists, this view is known as "hypothetical universalism." There were some delegates at the assemblies that wrote the Westminster Standards and the Canons of Dort who affirmed hypothetical universalism, and some advocates of the position today use that fact to argue that the position is not intended to be denied by either document. But others argue that the language of the two documents precludes the position and the small number of delegates who affirmed it were simply overruled. Since you mention Piper, I've seen him accused of holding the position of hypothetical universalism, but I don't think he does; I think he would align with what I outlined above instead.

I would encourage you to read the second head of doctrine in the Canons of Dort for the most definitive exposition of the doctrine often known as limited atonement.

Your question is about the statement that "Christ did not die for everyone" in the typical Reformed doctrine of Particular Redemption. Although you list 5 complex points in 5 paragraphs, the simplest way of cutting through confusion and clutter would be to express the typical Reformed doctrine in as straightforward and simple a manner as possible. This in itself, should aid understanding if it is used as a basis for building upon. First, I hope to clear up your main question, which is: “My main question is how this vague preposition "for" is meant to be interpreted in the sentence "Christ did not die for everyone"… In other words, they surely have to be willing to say that, in some senses, Christ did die for everyone.”

As Christ died for sinners (1 Timothy 1:15), and all humans are sinners, the question really is, ‘Which sinners did Christ die for, if not for all sinners?’ Jesus answered that in Mark 2:17. He was being sneered at by religious scribes and Pharisees for mixing with ‘publicans and sinners’, which group of ‘undesirables’ they studiously avoided. Jesus told them, “They that are whole have no need of the physician, but they that are sick. I came not to call the righteous, but sinners to repentance.” Therefore, although all people are sinners, those who consider themselves to be righteous will not be healed by Jesus. They will remain in their sin. His sacrificial death will not ‘cover’ them. Please bear that in mind as I now quote from a booklet dealing with what has come to be known as “The Five Points of Calvinism”, but I will confine this to the point of Particular Redemption / Limited Atonement.

First, this booklet shows what the Arminian point was that resulted in the Reformed view to counter it. It was point 3 which could be headed Universal Redemption, or General Atonement. This taught that Christ died to save all men; but only in a potential fashion. Christ’s death enabled God to pardon sinners, but only on condition that they believed. The counter-point could be called Limited Atonement i.e. Particular Redemption. Here is the explanation of what that means in this booklet:

First of all, they had asked, “Who is to be saved?” The answer was “Man”. But the Bible’s teaching with regard to man showed that man, in his natural state, is totally unable to save himself. Thus, we have the teaching of the Bible on man set under the general heading of Total Depravity, or Total Inability.

Secondly, as some men and women are undoubtedly save, then it must have been God Himself who had saved them in contra-distinction to the rest of mankind. This is Election: ‘That the purpose of God according to election might stand…’ (Rom. 9:11). However, this election only ‘marked the house to which salvation should travel,’ as Spurgeon puts it, and a full and perfect and satisfactory atonement was still required for the sins of the elect, so that God might be, not only a Saviour, but ‘a just God, and a Saviour.’ This atonement, as we all acknowledge, was accomplished through Christ’s voluntary submission to the death on the Cross where He suffered under the justice of this just God, and procured the salvation that he as Saviour had ordained. On the Cross, then – and, no doubt, we all accept this – Christ bore punishment, and procured salvation.

The question now arises, whose punishment did He bear, and whose salvation did He procure? There are three avenues along which we can travel with regard to this:

1. Christ died to save all men without distinction.

2. Christ died to save no one in particular.

3. Christ died to save a certain number.

The first view is that held by ‘Universalists’, namely, Christ died to save all men, and so, they very logically assume, all men will be saved. If Christ has paid the debt of sin, has saved, ransomed, given His life for all men, then all men will be saved. The second one is the ‘Arminian’ one, that Christ procured a potential salvation for all men. Christ died on the Cross, this view says, but although he paid the debt of our sins, his work on the Cross does not become effectual until man ‘decides for’ Christ and is thereby saved. The third view of the Atonement is the ‘Calvinistic’ one, and it says that Christ died positively and effectually to save a certain number of hell-deserving sinners on whom the Father had already set His free electing love. The Son pays the debt for these elect ones, makes satisfaction for them to the Father’s justice, and imputes His own righteousness to them so that they are complete in Him.

Christ’s death, then, could only have been for one of these three reasons: to save all; to save no one in particular; to save a particular number. The third view is that which is held by the Calvinist and is generally called limited atonement, or particular redemption. Christ died to save a particular number of sinners; that is, those ‘chosen in him before the foundation of the world’ (Eph. 1:4); those whom the Father had ‘given him out of the world’ (John 17:9); those for whom He Himself said He shed His blood: ‘This is my blood of the new covenant, which is shed for many, for the remission of sins’ (Matt 26:28). The Five Points of Calvinism, W.J. Seaton, pp 14-16, The Banner of Truth Trust, Edinburgh Scotland, 1984 reprint.

There is more on this point, mainly dealing with many scriptures that can be seen to support this view of limited atonement but I am not to here to argue the case for this – only to explain something of what this view actually states. I suggest that it is the word ‘to’ that is key, not the word ‘for’. What did Christ sacrificially die to do? Sort that out first, and then those for whom he died might become clearer. Perhaps this is what prevents clear understanding of this point?

Although you raise other related questions, I will leave it at this, because clarity about what the ‘Limited Atonement’ view says is needed first. Thereafter, progress might be made clearing up those related points. I trust this will be helpful, and just add another source that might be of help, namely, http://www.blogos.org/compellingtruth/calvinism-misconceptions.php