  The dynamics of interacting dark matter-dark energy models is characterized
through an interaction rate function quantifying the energy flow between these
dark sectors. In most of the interaction functions, the expansion rate Hubble
function is considered and sometimes it is argued that, as the interaction
function is a local property, the inclusion of the Hubble function may
influence the overall dynamics. This is the starting point of the present
article where we consider a very simple interacting cosmic scenario between
vacuum energy and the cold dark matter characterized by various interaction
functions originated from a general interaction function: $Q=
\Gamma\rho_{c}^{\alpha }\rho_{x}^{1-\alpha -\beta}(\rho_{c}+\rho_{x})^{\beta}$,
where $\rho_c$, $\rho_x$ are respectively the cold dark matter density and
vacuum energy density; $\alpha$, $\beta$ are real numbers and $\Gamma$ is the
coupling parameter with dimension equal to the dimension of the Hubble rate. We
investigate four distinct interacting cosmic scenarios and constrain them both
theoretically and observationally. Our analyses clearly reveal that the
interaction models should be carefully handled.
