Di Paolo on Extended Mind 3

[The] organism is linked with an external entity in a two-way interaction, creating a coupled system that can be seen as a cognitive system in its own right. All the components in the system play an active causal role, and they jointly govern behavior in the same sort of way that cognition usually does. If we remove the external component the system’s behavioral competence will drop, just as it would if we removed part of its brain. Our thesis is that this sort of coupled process counts equally well as a cognitive process, whether or not it is wholly in the head (Clark and Chalmers 1998, p. 7).

This seems to suggest a practical and operational way out of the problem. Perform a causal analysis of the coupled system and work out what processes contribute to cognitive performance. But of course, without a measure of relevance, a causal analysis will inevitably invite an unbounded spread of causes (e.g., isn’t oxygen obviously crucial for a human to solve math problems?). It is clear that what counts as cognitive (the second boundary) should be the measure that determines the relevance of the causal contribution of a given process. But this leads us again to the problem already stated. The only test of the cognitive offered by EM is whether we intuitively would call something cognitive were it to happen in the head. (Di Paolo, 2009, p. 10).

And, I have to agree that the way to separate what merely causally influences cognition and what constitutes cognition is to have a mark of the cognitive.  We need a theory of what distinguishes cognitive processes from non-cognitive processes.
Di Paolo, E. (2009). Extended life. Topoi, 28(1), 9-21.