  Extensions of the gravitational framework of Brans-Dicke (BD) are studied by
considering two different scenarios: i) `BD-$\Lambda$CDM', in which a rigid
cosmological constant, $\Lambda$, is included, thus constituting a BD version
of the vanilla concordance $\Lambda$CDM model (the current standard model of
cosmology with flat three-dimensional geometry), and ii) `BD-RVM', a
generalization of i) in which the vacuum energy density (VED),
$\rho_{\textrm{vac}}$, is a running quantity evolving with the square of the
Hubble rate: $\delta\rho_{\textrm{vac}}(H)\propto \nu\, m^2_{\textrm{Pl}}
(H^2-H_0^2)$ (with $|\nu|\ll 1$). This dynamical scenario is motivated by
recent studies of quantum field theory (QFT) in curved spacetime, which lead to
the running vacuum model (RVM). We solve the background as well as the
perturbation equations for each cosmological model and test their performance
against the modern wealth of cosmological data, namely a compilation of the
latest SNIa+$H(z)$+BAO+LSS+CMB observations. We utilize the AIC and DIC
statistical information criteria in order to determine if they can fit better
the observations than the concordance model. The two BD extensions are tested
by considering three different datasets. According to the AIC and DIC criteria,
both BD extensions i) and ii) are competitive, but the second one (the BD-RVM
scenario) is particularly favored when it is compared with the vanilla model.
This fact may indicate that the current observations favor a mild dynamical
evolution of the Newtonian coupling $G_N$ as well as of the VED. This is in
agreement with recent studies suggesting that the combination of these two
features can be favorable for a possible resolution of the $\sigma_8$ and $H_0$
tensions. In this work, we show that the Brans-Dicke theory with running vacuum
has the potential to alleviate the two tensions at the same time.
