We review the screening nature and many-body physics foundation of the van der Waals density functional (vdW-DF) method [ROPP 78, 066501 (2015)], a systematic approach to construct truly nonlocal exchange-correlation energy density functionals. To that end we define and focus on a class of consistent vdW-DF versions that adhere to the Lindhard screening logic of the full method formulation. The consistent-exchange vdW-DF-cx version [PRB 89, 035412 (2014)] and its spin extension [PRL 115, 136402 (2015)] represent the first examples of this class; In general, consistent vdW-DFs reflect a concerted expansion of a formal recast of the adiabatic-connection formula [PRB 90, 075148 (2014)], an exponential summation of contributions to the local-field response, and the Dyson equation. We argue that the screening emphasis is essential because the exchange-correlation energy reflects an effective electrodynamics set by a long-range interaction. Two consequences are that 1) there are, in principle, no wiggle room in how one balances exchange and correlation, for example, in vdW-DF-cx, and that 2) consistent vdW-DFs has a formal structure that allows it to incorporate vertex-correction effects, at least in the case of levels that experience recoil-less interactions (for example, near the Fermi surface). We explore the extent to which the strictly nonempirical vdW-DF-cx formulation can serve as a systematic extension of the constraint-based semilocal functionals. For validation, we provide new vdW-DF-cx results for metal surface energies and work functions that we compare to experiment, noting that vdW-DF-cx performs at least at the same level as the popular constraint-based semilocal functions. This is true even though vdW-DF-cx exclusively relies on input from quantum Monte Carlo and formal many-body physics theory to set both exchange and correlation. Finally, we use the screening insight to separate the vdW-DF nonlocal-correlation term into pure-vdW-interaction and local-field-susceptibility effects and present tools to compute and map the binding signatures of these mechanisms in isolation.