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 (STT) and general relativity with a scalar field either minimally or nonminimally coupled to matter. Higher-derivatives Horndeski and beyond Horndeski theories that at first sight do not look like STT 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 STT 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.