In fn #8, p. 212, of Radical Embodied Cognitive Science, Chemero writes,
"Adams and Aizawa (2008) argue that defenders of the sort of view of cognition that I am defending here need to give a definition of 'cognition'".
I don't think A&A ever mention a need to define "cognition" and at one point we were pretty explicit about not defining "cognition" ourselves. (See below the fold.) For A&A, giving a "mark of the cognitive" is not meant to be defining "cognition".
In fact, if one reads through the remainder of Chemero's footnote, one finds Chemero using something like the meta-theory A&A use to specify how they distinguish cognitive processes from non-cognitive processes.
In Chapters 3 and 4, we develop and defend in more detail our positive approach to the mark of the cognitive, namely, that cognitive processes differ from non-cognitive processes in terms of the kinds of mechanisms that operate on non-derived representations. We offer this as part of a theory of the cognitive, rather than as (part of) a definition of the term “cognitive.” We do not mean to stipulate that this is just what we mean by “cognition.” Nor do we mean to be offering an account of what “folk psychology” or common sense maintains about what cognition is. One consequence of offering a partial theory of the cognitive, rather than any of these other things, is that we can refine it only as far as (we take) the current evidence in cognitive psychology to warrant. (pp. 12-3)