We generalize the previously proposed running vacuum energy model by including a term proportional to ˙ H , in addition to the existing H 2 term. We show that the added degree of freedom is very constrained if both low redshift and high redshift data are taken into account. Best-fit models are undistinguishable from Λ CDM at the present time, but could be distinguished in the future with very accurate data at both low and high redshifts. We stress the formal analogy at the phenomenological level of the running vacuum models with recently proposed dark energy models based on the holographic or entropic point of view, where a combination of ˙ H and H 2 term is also present. However those particular entropic formulations which do not have a constant term in the Friedmann equations are not viable. The presence of this term is necessary in order to allow for a transition from a decelerated to an accelerated expansion. In contrast, the running vacuum models, both the original and the generalized one introduced here contain this constant term in a more natural way. Finally, important conceptual issues common to all these models are emphasized.