I think the key distinction between intrinsic and extrinsic motivation is not the awareness of the motive, but the locus of origin of the motive.
We are very much aware of intrinsic motives. For example: Hunger- Going without food for a long time makes us hungry, a state we are aware of, and we consume food. The act of consuming food when hungry is intrinsically motivated, and emerges from within us, without the effect of any external agent. However, Ryan and Deci's self-determination theory (SDT) did not mention (as far as I know) intrinsic motives as being connected to a physiological need. SDT's intrinsic motives were connected to psychological needs, in doing activities which were enjoyable in themselves, unconnected to an external reward.
As you have mentioned, we are also aware of our extrinsic motives, for example, motivation to work a 9-to-5 job to earn money to buy food which satiates hunger, a primary need.
My interpretation is that since extrinsically motivated activities are oriented towards an external goal and since "goals are cognitive
representations of what individuals are trying to accomplish and their purposes
or reasons for doing the task" (3), hence the authors are stating that extrinsic motives have cognitive representations.
My conflict is that granted, extrinsic motivation is goal-oriented but isn't intrinsic motivation also goal-oriented? For example, mastery in a task if often considered to be intrinsically motivated. Does that not mean the mastery=goal? As you will see from the table of motivational terms in (4) (see Garcia and Pintrich, 1996 in the table), intrinsic motives are also goals, according to many behavioural scientists. That would make intrinsic motives to also be cognitively represented.
In their discussion of the accessibility of motivational constructs, Murphy and Alexander really raise two issues; one concerns the consciousness or cognitive accessibility of motivational constructs and the other concerns the accuracy of individuals’ reports of their motivation. From an achievement goal theory perspective, the issue of cognitive consciousness or accessibility is not as major a problem as it may be for models that assume there are some ‘‘deeply-held, pervasive motives, needs, or drives’’ to use Murphy and Alexander terms (this issue). Goal theory assumes that goals are cognitive representations of what individuals are trying to accomplish and their purposes or reasons for doing the task. As such, they are inherently cognitive and assumed to be accessible by the individual; they are not unconscious motives as in psychodynamic theory, nor are they deeply held needs or motives as in some models of motivation (cf. Deci & Ryan, 1985; McClelland, Atkinson, Clark, & Lowell, 1953; Murray, 1938). In fact, the metaphor of ‘‘excavating’’ deeply held motives or needs and bringing them to the ‘‘surface’’ or into the ‘‘light’’ of day through various methods such as projective tests or intensive psychotherapy is not really applicable to achievement goal theory. In contrast, the discussion of goal constructs ﬂows from the general cognitive revolution in psychology and goals are assumed to be internal, cognitive representations or knowledge structures.
In conclusion, I don't understand why only extrinsic motives have been given a cognitive representation and not intrinsic motives, in the text.
I highly recommend you to go through (3) and (4), since this seems to be a matter of confusion regarding how motivational constructs are defined. There has been considerable debate regarding the definitions of motivational constructs and what concepts they encompass. The quoted line in the question seems to be a result of this very dilemma.
1) Psychology, Ciccarelli, fourth edition.
3) Pintrich, P. R. (2000). An achievement goal theory perspective on issues in motivation terminology, theory, and research. Contemporary educational psychology, 25(1), 92-104.
4) Murphy, P. K., & Alexander, P. A. (2000). A motivated exploration of motivation terminology. Contemporary educational psychology, 25(1), 3-53.