An issue with the classification of the scalar-tensor theories of gravity

In the bibliography a certain confusion arises in what regards to the classification of the gravitational theories into scalar-tensor theories and general relativity with a scalar field either minimally or non-minimally coupled to matter. Higher-derivatives Horndeski and beyond Horndeski theories that at first sight do not look like scalar-tensor theories only add to the confusion. 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. In this paper we propose a specific criterion for an unambiguous identification of scalar-tensor theories and discuss its impact on the conformal transformations issue. The present discussion carries not only pedagogical but also scientific interest since an incorrect classification of a given theory as a scalar-tensor theory of gravity may lead to conceptual issues and to the consequent misunderstanding of its physical implications.