As I understood it: C to K would be linear since they are always offset by the same amount, but 0°C is not 0°K. C to F is not linear since they are not offset by the same amount.
It seems to me C to K is affine as well. True, it's somewhat special in that the factor is exactly one, but if there's an offset it's not linear in the sense of the sentence under discussion.
That's not how it works. The definition of linear used in this context requires f(a+b) = f(a) + f(b) which is not the case with the C to K translation which is f(x) = x + 273.15 Only functions in the form y=mx+0 meet the criteria.