An issue with the classification of the most general scalar-tensor theories

In the present paper we discuss on the issue that arises when one tries to classify the gravitational theories into scalar-tensor theories or general relativity with a scalar field non-minimally coupled to matter. Despite that the issue might seem like a trivial one, some confusion might arise when dealing with higher-derivatives Horndeski theories that at first sight do not look like scalar-tensor theories. To further complicate things, the discussion on the physical equivalence of the different conformal frames in which a given scalar-tensor theory may be formulated, makes even harder to achieve a correct classification.