This could become a very complex discussion, Heather, so thanks for opening it as its own topic.

I'd tentatively suggest that a safe working label for your community, for now, is "a study group". Study groups have been critical pockets of learning - and inquiry - for thousands of years, and any learning theory should ultimately have something useful to say about them. To what degree your study group ends up acting as a community of inquiry is arguably something to be investigated, perhaps as an initial step towards helping it develop.

Some possible points that come to mind:

Note that I'm using the lower-case letters: a [lower-case] community of inquiry is essentially a phenomenon, a "spirit of communal inquiry" that may or may not emerge out of a group of people working together to investigate or learn something. Its philosophical roots (eg., Peirce) go back to communities of scientists, with the argument that we can only approach something like "scientific truth" through a deliberate collective process. Importantly, those scientists weren't led by teachers; they led themselves. The application to education came later.

Not every study group will have that spirit or commitment to collective inquiry. Some study groups might live their lives happily as a way to share notes more efficiently or for social support only - in other words, not the full range of presences.

On the other hand, the Community of Inquiry [upper-case] is a framework and set of tools to design a more deliberate community of inquiry [lower-case], integrating all three presences, without having to leave so much up to chance. So "ad hoc" and "CoI" [upper-case] might not sit as comfortably together. A design implies some sort of designer, ie., a leader, and some form of teaching presence. But that doesn't mean the design, leadership, and teaching will all be concentrated into one person; in fact, the "spirit of communal inquiry" would suggest it shouldn't be.

So the distinction between "a community of inquiry" (spirit) and "the Community of Inquiry" (structure) might become important, and where your study group falls - or could fall - between the two might depend on how "ad hoc, undesigned, and teacherless" it actually turns out to be. It'll be interesting to see what you find out.

Thank you for your insights, Dan! As I learn more about CoI and understand the constructs that define the presences, I'm noticing areas of overlap and harmony with other theories. I am working out where to blend edges versus draw lines. After a while, it is a bit like sampling too many perfumes. They all have some things in common, but have different names and essential elements. It reminds me of my biology background... Do we "lump" or "split" when classifying organisms? Focus more on grouping by similar characteristics versus dividing by distinguishing differences? I admit I'm a novice at learning theories... Trying to put together puzzles with infinite numbers of pieces, undefined corners, and a moving picture on the box. :-)

Yes - because learning is so complex and so intangible, we'll likely never be able to approach it directly, so it might be useful to see each theory as trying to approach it from a different perspective, with its own beliefs, values, and goals.

The CoI framework, for example, could be interpreted as trying to integrate three different perspectives on the central idea of an "educational experience", and how these perspectives overlap reveals additional perspectives.

You can see more from some perspectives than from others, but none of them captures the full complexity of that central idea. Each perspective can be defined precisely, but you can shift your point of view smoothly from one to the next. If you share the same perspective as those around you (say, your committee), then you can often have a richer conversation with them about it - usually a good way to get started. Introducing a new perspective can potentially lead to a richer understanding in the end, if you're clear on what your perspective is, including your beliefs, your values, and your goals.

If you haven't already, you might look up "paragogy" (sometimes called "peeragogy") - not necessarily as yet another alternative to the CoI but as an additional perspective on what the CoI and many others are trying to do.