Since this is a meta discussion, I'll provide an answer as well, specifically addressing the issue of clarity.
It seems to me that there is a significant issue here on CV with respect to general <=> specific questions. My understanding of SE sites is that we want to accumulate useful information for a general audience. Thus it makes more sense to formulate questions in ways that make them as relevant for people as possible.
A good but possibly silly analogy would be as follows; say I want to know how to throw a baseball, because I don't know how to throw a ball. I could ask the question as "how can I throw a baseball on a grass patch behind my house?" or I can ask something like "how to throw a hand sized ball outdoors?". The second alternative makes it immediately more relevant for wider audience, those who might want to throw a tennis ball, or a lacrosse ball, or a softball... I think putting the effort to generalize and make the question relevant for many people should not be penalized here.
The second issue is how much details one has the liberty to give. Now in this case the details are neither government nor corporate secrets. However they are research results and it is rather tricky what I am allowed to say and not say, especially considering the figures. There are novelty concerns for both academic and IP settings.
However, not all the details are relevant to the question, for example the exact definitions of $a_i$, $b_i$ and $c_i$, are not relevant to what I was trying to ask, IMHO. They can be increasing doses of a medication each, they can be measurements with increasing amount of time, amount of food or dollars received for a task in a behavioral experiment... The point is that $a_4$ is more $a$ than $a_3$. The underlying assumptions is that $a_1$ to $a_4$ are increasing level of stimulus of a specific kind, analogous with $b_1$ to $b_4$ and $c_1$ to $c_3$.
The idea is to find variables that respond to $a$ or $b$ or $c$ and not to combinations of them. That's what I meant with classification or categorization. So yes it is supervised, if you in advance specify which clusters to find. As an alternative I tried to automate the process by adopting an unsupervised method to see if it would pick up the same clusters. They did not, as I have explained in the question body.
The issue here is that people seem to get stuck on details that are not going to be relevant for them. I am fairly certain that practically nobody here would be able to relate to the kind of processes $a$, $b$ and $c$ refer to. That's because I barely grasp how they work myself. :)
I tried to explain the data as best as I can, and I am still willing to explain bits if someone is interested in the specifics of the problem at hand. At the end of the day I am looking for leads to solve a problem I have, I don't want to outsource my job to someone else. I think it's the more respectable thing to do.