One way to characterize the difference between a presbyterian and a semi-congregationalist view of church government is to say that in the presbyterian view, the de jure church is inherently indivisible, whereas in the semi-congregationalist view, the de jure church is not inherently indivisible.
There is one Body of Christ in the world, and it is a sin to sever that body. Unity between true churches is a moral mandate, not an optional thing or a thing to be preserved only in ideal circumstances. Christians who are formally members of the church are bound in duty to be in unity with other formally recognized Christians. De jure elders are duty-bound to function collegially as part of a universal eldership over the entire church. When churches are separate from each other (as they are in multiple denominations), the members of one church are not in formal communion with members of the other church, and the elders and courts of one church are not united in mutually-binding councils with elders and courts in the other church. The churches are rejecting each others' de jure legitimacy and authority. Presbyterianism recognizes the biblical requirement of unity, and accordingly it maintains that there is an absolute duty of unity between all true de jure churches and no room for multiple de jure denominations. In other words, presbyterianism recognizes the inherent unity and indivisibility of the catholic church.
Semi-congregationalism, on the other hand, sometimes holds that denominational unity is not a required thing, or if it is required in some sense it is only required prima facie--when the circumstances are right. Semi-congregationalists don't hold that the de jure church is inherently and intrinsically, by nature and moral right, indivisible. It can indeed exist, and sometimes ought to exist, in multiple de jure denominations. This does not imply a mutual rejection of each others' de jure legitimacy and authority, because it is considered to be perfectly possible for there to be multiple, independent formal Bodies of Christ.
The catholic church is inherently indivisible in all its aspects. The catholic church invisible is indivisible, as all elect and regenerate Christians are necessarily united in one Spirit. The visible catholic church de facto is indivisible, as all Christians are duty-bound to maintain fellowship with each other informally to the extent that they informally recognize each other as brothers and sisters in Christ. And the visible catholic church de jure is indivisible, as all de jure true churches have an absolute duty always to maintain formal unity with each other (as, for example, manifested in the recognized ability and at least occasional duty to come together in binding presbyteries, synods, etc.). Presbyterianism recognizes, acknowledges, and practices this biblical unity, while semi-congregationalism, to some degree or another, gives it up.
For more, see here, here, here, and in general here.
UPDATE 9/3/13: An illustration of the difference between presbyterianism and semi-congregationalism regarding the unity and indivisibility of the church can be seen in how they deal with doctrinal and practical errors in the church.
When a church falls into doctrinal or practical error that is contrary to the Word of God, the presbyterian response is that the catholic church will discipline that church (by means of whatever council is appropriate). If the church does not repent, it will be cut off from the de jure catholic body, having its legitimacy and authority revoked until it repents. Thus, the unity and indivisibility of the de jure catholic church is preserved. There has been a schism from the body, but not a schism within the body, speaking de jure (while also recognizing that there has been a schism within the de facto Body of Christ--the de facto body retains its de facto indivisibility because there can still be informal fellowship, but the church that is cut off has lots its de jure, formal connection with the rest of the de jure church).
In a semi-congregationalist system, however, what often happens is that the churches separate from each other while still recognizing each other as true de jure churches. Instead of cutting the erring church off from the body until it repents, the semi-congregationalist way is often to simply divide the de jure catholic church itself into pieces, thus illustrating its deviation from the biblical principle of the indivisibility and mandated unity of the church.
UPDATE 9/3/13: Let me make one brief clarification: It seems theoretically possible that a situation could arise in which two churches split from each other, both of which are exactly equally right and equally wrong in their convictions/practices and both which have exactly identical historical and doctrinal claims to a right to separate existence. In this case, each of the denominations considers itself alone to be the de jure church, because it holds itself to be not guilty of doctrinal or practical error in connection with the division. A third party, however, may perceive the errors on both sides and come to the conclusion that both have an equal claim to separate existence. Both of the churches, from the perspective of the third party, have embraced some error that is a cause of legitimate concern to the other side, so both sides have acted rightly in trying to discipline the error of the other side, but both sides also are schismatic because they have embraced errors requiring discipline. The ideal response to this situation would be for a party to break off and form a third denomination which preserves the truths on both sides, and if that were to happen, the third denomination would alone possess de jure authority. However, if the perceptive third party observing what is really going on is unable to bring about the formation of a third denomination in an appropriate manner, he may be forced for the time being to join one of the two denominations, declaring them both to have de jure authority. In this sense, we could say that it is possible for the de jure church to be divided. However, note that this is a sinful and not an appropriate situation. It is never, ever justifiable for one de jure church to be separate from another de jure church. Unity is an absolute moral mandate. That is what I mean when I say that the de jure church is indivisible: Its essence involves unity, and so it can never be legitimate to divide it. If one tries to divide it, one is always acting contrary to the commands of God regarding the essential nature of the church. That is the presbyterian view. Semi-congregationalism, on the other hand, holds that it is sometimes permissible for a de jure church to be divided from another de jure church. That is how the two views are different (at least concerning this issue).
St Edmund and Abingdon
6 days ago