The whole article is now online in PDF at the Gospel Coalition site.
Here's a summary of what Carson attempts to do in this chapter from 1996:
Historically, evangelicals have been concerned to preserve and promulgate the gospel. But precisely what is this gospel? All sides recognize that it is ‘good news’ in some sense. But what is the content of this good news?I encourage you to read the whole thing. Regarding the relationship betweenthe "gospel of Jesus Christ" and the "gospel of the kingdom" (which some separate), Carson observes: "The point is that all of God’s sovereignty is mediated through the resurrected and exalted Jesus (1 Corinthians 15)." Further, "King Jesus 'reigns from the cross,' as early patristic writers put it." Therefore, "There is no conflict between these two designations ["gospel of the kingdom," "gospel concerning God's Son"], and both drive us to focus on Jesus Christ, the Son of God, the King, whose utterly extraordinary mission was to die the odious death of an accursed wretch, in fulfilment of OT patterns and pictures and prophecies of sacrifice."
Although it is worth saying something about the ‘gospel’ words in the NT, the issue cannot be answered by mere word studies. Some NT books, eg John’s Gospel, never use the word ‘gospel’, even though from a thematic perspective they obviously have as much ‘good news’ to tell as books that abound in the ‘gospel’ word-group. In the pages that follow, therefore, after some observations on the relevant words, I outline some of the broader considerations that must be taken on board if we are to grasp what the biblical gospel is. And, finally, I outline the primacy of the gospel in all Christian thought and mission over against competitors and would-be usurpers.
By the way, a good exercise for everyone would be to attempt to write a single paragraph putting the gospel into the storyline of Scripture. Here's how Carson does it:
Thus the gospel is integrally tied to the Bible’s story-line. Indeed, it is incomprehensible without understanding that story-line. God is the sovereign, transcendent and personal God who has made the universe, including us, his image-bearers. Our misery lies in our rebellion, our alienation from God, which, despite his forbearance, attracts his implacable wrath. But God, precisely because love is of the very essence of his character, takes the initiative and prepared for the coming of his own Son by raising up a people who, by covenantal stipulations, temple worship, systems of sacrifice and of priesthood, by kings and by prophets, are taught something of what God is planning and what he expects. In the fullness of time his Son comes and takes on human nature. He comes not, in the first instance, to judge but to save: he dies the death of his people, rises from the grave and, in returning to his heavenly Father, bequeaths the Holy Spirit as the down payment and guarantee of the ultimate gift he has secured for them—an eternity of bliss in the presence of God himself, in a new heaven and a new earth, the home of righteousness. The only alternative is to be shut out from the presence of this God forever, in the torments of hell. What men and women must do, before it is too late, is repent and trust Christ; the alternative is to disobey the gospel (Romans 10:16; 2 Thessalonians 1:8; 1 Peter 4:17).