Oops. So, A&A did talk about a definition of the cognitive

Doing a word search over Bounds I found the following:

One way to think about our strategy for demarcating cognitive from non-cognitive processes is to begin with paradigms of cognitive processing, those involving normal humans.  We have been drawing on features of human cognition as a first step towards demarcating the cognitive from the non-cognitive.  But, surely the category of the cognitive encompasses more than this.  Surely a definition of the cognitive exclusively in terms of normal human cognition is too parochial. (Bounds, p. 70).

It would have been better to write "account" rather than "definition" in that last sentence.

So, against this.  There is no use of "definition" in Adams and Aizawa, (2001), and the following disclaimers in Bounds.

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.”   (Adams & Aizawa, 2008, pp. 12-13) 

Here we think it is perfectly reasonable for us to stand by the view that these Martian representational states are not cognitive states.  We have a theory of what cognition involves.  The Martians in Clark’s thought experiment do not satisfy the conditions of that theory.  So we must either reject the hypothesis that the Martians have cognitive processing or the hypothesis that cognition involves non-derived representations.  Why can we not rationally choose to stand by our theory?  Our theory is an empirical conjecture about the nature of cognition, not a definition of cognition.  Thus, future scientific developments could undermine our theory and force revisions.  Or, our theory could turn out to be so successful and well-confirmed that we determine that Martians are not cognizers. (Adams & Aizawa, 2008, p. 49)

In this chapter we have offered an empirical hypothesis concerning what all cognitive processes have in common, namely, that they all involve non-derived representations.  We do not take this to be part of a definition of the cognitive.  Nor do we mean to stipulate what we shall mean by the word “cognitive.”  (Adams & Aizawa, 2008, p. 55)

Evidently, the dispute must be joined by a substantive theory of the cognitive.  This is why we offer the conjecture that cognitive processes involve non-derived representations that are embedded within (largely unknown) cognitive mechanisms.  This is not a definition of the cognitive, let alone a stipulative definition of the cognitive.  It is a theory that we think is implicitly at work in a lot of cognitive psychological research.  (Adams & Aizawa, 2008, p. 84)

So, there is a little infelicity.