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: $Q= Gammarho_{c}^{alpha }rho_{x}^{1-alpha -beta}(rho_{c}+rho_{x})^{beta}$, 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.
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