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 $3-3.5sigma$ 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
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