J. Evans’ 2008 paper reviews a number of proposed dual-processing descriptions of cognition. (see citation). He finds some stable divisions in the theories but a number of conflicting ones also. Calling the dual processing system 1 and system 2 appears to him to give a mistaken impression of how cognition works; he feels that processes (not locations, contents, parallel systems etc.) is the key difference and uses the terms type 1 and type 2.
A core type 2 is fairly easy to define slow, sequential, limited capacity, associated with working memory and conscious experience. Around this core, various theories diverge. Type 1 seems almost impossible to define at present with much agreement, except that it is not type 2. It may not be (probably isn’t) a single type of process. I look for types 1a, 1b etc. in future.
Here is part of the concluding discussion in the paper:
Although dual-process theories have been around in cognitive and social psychology for 30 years and more, it is only within the past 10 years or so that the terms System 1 and System 2 have come into common use. However, close inspection of the evidence suggests that generic dual-system theory is currently oversimplified and misleading. In particular, (a) it is not possible coherently to link together all the attributes associated with Systems 1 and 2 and (b) there are at least two quite distinct forms of dual-process theory to be found in these various literatures that can not readily be mapped on to each other. We might be better off talking about type 1 and type 2 processes since all theories seem to contrast fast, automatic, or unconscious processes with those that are slow, effortful, and conscious. However, it would then be helpful to have some clear basis for this distinction. My suggestion is that type 2 processes are those that require access to a single, capacity-limited central working memory resource, while type 1 processes do not require such access. This implies that the core features of type 2 processes are that they are slow, sequential, and capacity limited. The last feature implies also that their functioning will correlate with individual differences in cognitive capacity and be disrupted by concurrent working memory load. …
However, other proposed features of System 2 in the generic theory do not immediately follow from this definition of type 2 processes, for example, the proposal that such processes are uniquely human or associated with decontextualized thought or rule-based reasoning.
The problem with this distinction is that type 1 processes then simply refer to any processes in the mind that can operate automatically without occupying working memory space. As already indicated, there are a number of different kinds of such implicit processes. We may have innate cognitive modules with encapsulated processes for perception, attention, language processing, and so on. We appear to have an associative learning system that implicitly acquires knowledge of the world in a form similar to weights in neural networks; the knowledge cannot be called to mind as explicit knowledge, but it can directly affect our behavior. We have habitual and automated behavior patterns that once required conscious type 2 effort but seem to have become type 1 with practice and experience. We also have powerful pragmatic processes that rapidly identify and retrieve explicit knowledge for conscious processing. Type 2 processing requires supporting type 1 processes to supply a continuous stream of relevant content into working memory. If there are indeed multiple kinds of type 1 processes, then it is to be expected that psychologists will have developed different kinds of dual-process theories, which seems to be the case. …
In short, my conclusion is that although dual-process theories enjoy good empirical support in a number of fields of psychology, the superficially attractive notion that they are all related to the same underlying two systems of cognition is probably mistaken, at least in the way that Systems 1 and 2 are being defined in the current literatures. For example, it is almost certainly wrong to think of System 1 as one system, all of which is old and shared with other animals. Equally, it is probably a mistake to think of System 2 as the conscious mind, all of whose processes are slow and sequential. If there is a second system, distinctively human, involving working memory and neurologically distinct structures, it does not follow that all of its workings are conscious and controlled. It is perfectly possible that one system operates entirely with type 1 processes and that the other includes a mixture of type 1 and type 2 processes, the latter being linked to the use of working memory, which this system uses – among other resources. Such a proposal could resolve the conflict between evidence for dual systems on the one hand with the proposals of different dual-process theorists on the other.
Dual-processes of cognition seems the wrong description of what is likely to be a very complex mixture of types and combinations of types. Again reality refuses to be neatly divided into just two parts.
J. Evans (2008). Dual-Processing Accounts of Reasoning, Judgement, and Social Cognition Annu. Rev. Psychol. (59), 255-278