  There is no unique way to describe the dark energy-dark matter interaction,
as we have little information about the nature and dynamics of the dark sector.
Hence, in many of the phenomenological dark matter fluid interaction models in
the literature, the interaction strength $Q_{\nu}$ in the dark sector is
introduced by hand. Demanding that the interaction strength $Q_{\nu}$ in the
dark sector must have a field theory description, we obtain a unique form of
interaction strength. We show the equivalence between the fields and fluids for
the $f(R,\chi)$ model where $f$ is an arbitrary, smooth function of $R$ and
classical scalar field $\chi$, which represents dark matter. Up to first order
in perturbations, we show that the one-to-one mapping between the
\emph{classical} field theory description and the phenomenological fluid
description of interacting dark energy and dark matter exists \emph{only} for
this unique form of interaction. We then classify the interacting dark energy
models considered in the literature into two categories based on the
field-theoretic description. We introduce a novel autonomous system and its
stability analysis for the general interacting dark sector. We show that the
dark-energy dominated epoch occurs earlier than the non-interacting systems for
a specific scalar field potential and a range of coupling strengths.
