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.
