Then your definition of an increase in information is irrelevant to how biology actually works. Substitution mutations, those that change one base to another DNA base, do change phenotype and are responsible for the difference in phenotype seen between species.
You have effectively argued yourself out of the conversation by using a definition for information that has no biological relevance.
But his example also changed the phenotype. It's just that the new 'organism' did not contain more 'information' that the previous one, as information is here arbitrarily defined.
I would agree that the analogy is not really relevant, but not because an SNP mutation does increase information. Does it?