If this article is characteristic of Warfield's thought in this area, then I can safely say that at least in the areas of church authority and unity, Warfield was no presbyterian but was rather a congregationalist (or at best a semi-congregationalist--but is this really different?). He emphatically denies ecclesiastical authority outside of the local churches, which is the historic hallmark of independency as opposed to presbyterianism:
It is important to observe, however, that this unity was not organic, in the special sense of that word which would imply that it was founded on the inclusion of the whole Church under one universal government. The absence of such an organization is obvious on the face of the New Testament record, nor do its pages contain any clear promise of or prominent provision for it for the future. The churches are all organized locally, but no external bonds bind them together, except as this was here and there supplied to certain groups of churches by the common authority over them of the same apostolical founders. No central authority ruled over the whole Church. It is perfectly obvious that Jerusalem exercised no domination over Antioch, Antioch none over the churches founded by its missionaries. Nor were the churches associated in a common dominion of the whole over all the parts. Even in the next generation the most powerful lever Rome could bring to bear on Corinth was entreaty and advice. The apostles went forth to evangelize the world, not to rule it; they divided the work among themselves, and did not seek to control it as a 'college'; they delegated their individual authority to the local officers and founded no dynasty, whether individual or collegiate.
Presbyterians would agree that "no central authority ruled over the whole Church," in the sense of a single person (like the Pope) or some group of people above the ordinary officers of the local churches (such as cardinals or archbishops). But Presbyterians would also equally insist that there is indeed another kind of central authority ruling over the church--precisely one of the kinds, in fact, that Warfield explicitly excludes when he says, "[n]or were the churches associated in a common dominion of the whole over all the parts." The apostles, Warfield says, "delegated their individual authority to the local officers and founded no dynasty, whether individual or collegiate." But this is not true, for according to Presbyterianism, the elders of the local churches do not function independently but are part of a college of elders ruling over the entire church. This is the foundation for the authority of presbyteries and synods or councils.
1. For the better government, and further edification of the Church, there ought to be such assemblies as are commonly called synods or councils.
3. It belongeth to synods and councils, ministerially to determine controversies of faith and cases of conscience, to set down rules and directions for the better ordering of the public worship of God and government of His Church; to receive complaints in cases of maladministration, and authoritatively to determine the same: which decrees and determinations, if consonant to the Word of God, are to be received with reverence and submission; not only for their agreement with the Word, but also for the power whereby they are made, as being an ordinance of God appointed thereunto in His Word. (Westminster Confession of Faith, Chapter 31)
CHRIST hath instituted a government, and governors ecclesiastical in the church: to that purpose, the apostles did immediately receive the keys from the hand of Jesus Christ, and did use and exercise them in all the churches of the world upon all occasions.
And Christ hath since continually furnished some in his church with gifts of government, and with commission to execute the same, when called thereunto.
It is lawful, and agreeable to the word of God, that the church be governed by several sorts of assemblies, which are congregational, classical, and synodical. . . .
IT is lawful, and agreeable to the word of God, that the several assemblies before mentioned have power to convent, and call before them, any person within their several bounds, whom the ecclesiastical business which is before them doth concern.
They have power to hear and determine such causes and differences as do orderly come before them.
It is lawful, and agreeable to the word of God, that all the said assemblies have some power to dispense church-censures. . . .
Synodical assemblies may lawfully be of several sorts, as provincial, national, and oecumenical. (Form of Presbyterial Church Government)
In short, according to Presbyterianism, the presbyterate of the church does not function simply in isolated factions, over local congregations, with (as Warfield puts it) "no external bonds" to "bind them together." Rather, the presbyterate is collegial in nature, as individual elders function as parts of congregational sessions, sessions function in submission to presbyteries, and presbyteries function in submission to larger synods or councils. There is no office above that of the the local elder (teaching or ruling), but the elders collectively exercise an authority over each other and have an obligation to submit to each other, and larger groups of elders have authority over individual elders or smaller groups of elders. As Charles Hodge sums it up, "The Presbyterian doctrine on this subject is, that the Church is one in such a sense that a smaller part is subject to a larger, and the larger to the whole."
It is astonishing that nowhere in Warfield's article does he even mention a passage of Scripture which clearly overturns his entire congregationalist mentality, and which is constantly on the lips of true presbyterians when they contend for the biblical view of church government. I refer, of course, to the narrative of the Jerusalem Council in Acts 15. This passage, among many others, is a great bane of congregationalists, for it shows the church doing exactly what they want to say the church is never to do--exercising collegial authority binding on the whole catholic church.
Warfield makes clear throughout his article that one of his main concerns is a false, latitudinarian idea of unity in which there is to be organizational unity among Christian at the expense of holding on to the whole counsel of God's Word. And he is quite right to warn us of this idea. But in avoiding this extreme, he seems to have gone to one on the other side and simply jettisoned the whole idea of organizational, governmental unity in the visible church. The truth is that we need not just unity, and not just truth, but unity and truth and unity in the truth.
Here are just a couple more quotations from Warfield's article, from towards the end of it where he is summing up some of the consequences of his principles:
1. We are not to seek it [that is, church unity] in the inclusion of all Christians in one organization and under one government. A story is told of a man who, wishing a swarm of bees, caught every bee that visited his flowers and enclosed them together in a box, only to find the difference between an aggregation and a hive. We cannot produce unity by building a great house over a divided family. Different denominations have a similar right to exist with separate congregations, and may be justified on like grounds. . . .
The true pathway seems, then, to lead us as our present duty to:
1. Hearty recognition of all Christians as members of the body of Christ, and of all denominations which preach the gospel of Christ as sections of this one body.
2. Hearty and unwavering testimony to all God's truth known to us, as the truth of God to be confessed by all his people.
3.Co-operation in all good works as brethren.
4.Formal federation of denominations for prosecuting tasks common to the federated bodies, so far as such federation involves no sacrifice of principle or testimony.
Contrary to Warfield, different denominations do not have a right to exist similar to that of separate congregations. Separate congregations are lawful and pragmatic divisions of the one church into local units for the better functioning and fellowship of the church (see the discussion of this in the Form of Presbyterial Church Government, linked to above). However, these distinct congregations have an obligation to function in visible and governmental unity with each other in a presbyterial system--under sessions, presbyteries, and higher synods. The whole idea of separate denominations, on the other hand, involves the idea of distinct churches or groups of churches refusing to submit to each other in formal, governmental unity, and this idea is inherently schismatic. Divided denominations may recognize each other as de facto parts of the visible church, but the very idea of governmental division implies (at least assuming a consistent presbyterian system) that the divided denominations do not recognize each other as parts of the visible church de jure.
For a brief biblical case for presbyterian church government, see here. For an examination of the Westminster Standards as they speak to the unity and collegial government of the church, see here. For more, see here.
UPDATE 2/3/15: Another concrete way of getting at how un-presbyterian Warfield's view is is to note that his view does not allow for any binding church assemblies beyond (at best) the level of presbytery. He says, "[t]he churches are all organized locally, but no external bonds bind them together, except as this was here and there supplied to certain groups of churches by the common authority over them of the same apostolical founders." He never (in this article, at least) clarifies whether by "locally" he is referring to the level of the local congregation or the level of the local presbytery. My guess would be the former, because in later Presbyterianism presbyteries in general seem to get a lot less local than they used to be. They seem almost to take over the role previously assigned to the "provincial synod."
Either way, Warfield's view does not recognize a presbyterial structure beyond the presbytery level. In the OPC, for example, there are congregational sessions, presbyteries, and the General Assembly. In the FPCS, there are sessions, presbyteries, and the Synod (with the idea that there would normally be a higher General Assembly if the church was larger). Warfield's view declares illegitimate and without authority the Synod and the General Assembly, because larger church assemblies, in his view, have no authority to rule over smaller church groups. So not only does Warfield deny the overall governmental unity of the visible church; his view also removes levels of governmental unity that are almost universally practiced in modern Presbyterian denominations. Boettner's view amounts to the same thing.