  We present a full-fledged analysis of Brans-Dicke cosmology with a
cosmological constant and cold dark matter (BD-$\Lambda$CDM for short). We
extend the scenarios where the current cosmological value of the BD-field is
restricted by the local astrophysical domain to scenarios where that value is
fixed only by the cosmological observations, which should be more natural in
view of the possible existence of local screening mechanims. Our analysis
includes both the background and perturbations equations in different gauges.
We find that the BD-$\Lambda$CDM is favored by the overall cosmological data as
compared to the concordance GR-$\Lambda$CDM model, namely data on distant
supernovae, cosmic chronometers, local measurements of the Hubble parameter,
baryonic acoustic oscillations, Large-Scale Structure formation and the cosmic
microwave background under full Planck 2018 CMB likelihood. We also test the
impact of Strong and Weak-Lensing data on our results, which can be
significant. We find that the BD-$\Lambda$CDM can mimic effective quintessence
with a significance of about $3-3.5\sigma$ c.l. (depending on the lensing
datasets). The fact that the BD-$\Lambda$CDM behaves effectively as a Running
Vacuum Model (RVM) when viewed from the GR perspective helps to alleviate some
of the existing tensions with the data, such as the $\sigma_8$ excess predicted
by GR-$\Lambda$CDM. On the other hand, the BD-$\Lambda$CDM model has a crucial
bearing on the acute $H_0$-tension with the local measurements, which is
rendered virtually harmless owing to the small increase of the effective value
of the gravitational constant with the expansion. The simultaneous alleviation
of the two tensions is a most remarkable feature of BD-gravity with a
cosmological constant in the light of the current observations, and hence goes
in support of BD-$\Lambda$CDM against GR-$\Lambda$CDM
