Is Allen Carr's "easy way to stop smoking" a form of cognitive therapy?
Cognitive therapy
The driving principle behind cognitive therapy is that mental problems are caused not by any 'objective' qualities of the events that we experience, but our perception and interpretation of those events. By changing these perceptions and interpretations, we can alleviate or solve those problems (Beck, 1970).
In this sense, Carr's method can be seen as consistent with cognitive therapy, but I would not go so far as to call it a form of cognitive therapy without peer review and validation that does not seem to exist.
Cognitive behavioral therapy
The difference between Burns' and Carr's methods can be explained by the fact that Burns is a cognitive behavioral therapist (CBT). This is a form of therapy which incorporates and expands on the principles of cognitive therapy, by acknowledging that not all of our perceptions and interpretations are under rational or conscious control. It therefore incorporates both cognitive and behavioral elements. An introduction to and meta-review of CBT's efficacy compared to therapy forms like cognitive therapy can be found in Butler et al. (2006).
Validity of Carr's method
Carr's work does not appear to have been directly referenced, at least not in any readily discoverable form, in the psychological literature. This may be because his method is actually not new, but seems to have been loosely based on a mix of Beck's ideas and Albert Ellis' stoicism-inspired rational therapy, so it should not necessarily be taken to mean that the method is nonsense; Carr may simply be popularizing established methods.
References

Beck, A. T. (1970). Cognitive therapy: Nature and relation to behavior therapy. Behavior therapy, 1(2), 184-200.
Butler, A. C., Chapman, J. E., Forman, E. M., &amp; Beck, A. T. (2006). The empirical status of cognitive-behavioral therapy: a review of meta-analyses. Clinical psychology review, 26(1), 17-31.