One way to distinguish between independency and presbyterianism is say that independency recognizes no authority beyond the local congregation or between local congregations, while presbyterianism does. In independency, the elders of one particular congregation (if there are any) have authority in the congregation, but they have no authority that elders or members in other congregations are bound to recognize and submit to. In presbyterianism, by contrast, elders in one congregation do have authority that elders and members in other congregations are bound to recognize. From thence arises the authority of presbyteries and higher synods. American Presbyterian theologian Charles Hodge put it well: "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."
There is no real non-arbitrary stopping point between independency and presbyterianism. I have sometimes spoken of "semi-independency" or "semi-congregationalism" as an in-between point of view, but really it is not. I have described semi-congregationalism as the view that churches can be united together in "clumps" (denominations) which hold a presbyterian structure of unity and authority within themselves, while the clumps themselves are independent from each other. But really, if there is no requirement for the clumps to recognize each others' authority, there can be no non-arbitrary reason to think that the churches within each clump are required, intrinsically, to recognize each others' authority either.
Consider this scenario: Denomination A holds to a presbyterian structure within itself. It recognizes the legal legitimacy and authority of Denomination B, but it holds that it has no obligation to be united to Denomination B in a mutually-binding presbyterian structure. The implication of this is that there is nothing about the authority of the elders in Denomination B which requires the recognition and submission of the elders in Denomination A. But if this is the case, how can there be anything in the authority of the individual sessions and presbyteries of Denomination A which would require the recognition and submission of other sessions and presbyteries within Denomination A? We can't have it both ways. Either the authority of elders requires submission from other elders, in which case Denomination A's recognition of authority in Denomination B will morally require Denomination A to join with Denomination B in a presbyterian structure (in other words, they will no longer be two independent denominations), or the authority of elders does not require submission from other elders, in which case the individual sessions and presbyteries within Denomination A will have no intrinsic moral obligation to be united to the other sessions or presbyteries within Denomination A--but this is nothing other than pure independency. Independency has no problem with churches voluntarily working together; what defines it in distinction from presbyterianism (at least in this regard) is that it does not recognize a moral requirement for churches to function in mutual submission to each other in a presbyterian system. It would be completely arbitrary for Denomination A to say that its member sessions have an intrinsic moral obligation to be united with other sessions within Denomination A but not with sessions in Denomination B. So the only conclusion we can draw is that there is no non-arbitrary stopping point between a full presbyterianism which is incompatible with denominationalism and pure independency.
For more, see here and here.
UPDATE 11/24/14: Given the above, here is something you can do to call out independent thinking in a particular church session: Ask the session, "Why do you submit to your presbytery? What is the basis of the authority they have over you? Is the basis your own voluntary decision to grant them authority over you, or do they have authority simply by virtue of containing legitimate officers and courts of the church of Christ within the same region in which you live?" If the answer is the former, then the session acknowledges no real intrinsic authority to be possessed by the presbytery as such, in which case the session's view is independency. If the answer is the latter, then you can ask, "What about the sessions of other denominations in the same region? Wouldn't they have authority too, to join together with your denomination's sessions in the meetings of presbytery? On what basis do you exclude them? And how do you avoid uniting in a common general council with all the other churches in the region, in the nation, and in the world, if church courts and officers, by virtue of their intrinsic authority, are required to be in mutual submission to each other?"
'An elephant in its bath'
4 weeks ago