  We study a model of weakly ionized, protostellar accretion discs that are
threaded by a large-scale, ordered magnetic field and power a centrifugally
driven wind. We consider the limiting case where the wind is the main
repository of the excess disc angular momentum and generalize the radially
localized disc model of Wardle & K\"onigl (1993), which focussed on the
ambipolar diffusion regime, to other field diffusivity regimes, notably Hall
and Ohm. We present a general formulation of the problem for nearly Keplerian,
vertically isothermal discs using both the conductivity-tensor and the
multi-fluid approaches and simplify it to a normalized system of ordinary
differential equations in the vertical space coordinate. We determine the
relevant parameters of the problem and investigate, using the
vertical-hydrostatic-equilibrium approximation and other simplifications, the
parameter constraints on physically viable solutions for discs in which the
neutral particles are dynamically well coupled to the field already at the
midplane. When the charged particles constitute a two-component ion--electron
plasma one can identify four distinct sub-regimes in the parameter domain where
the Hall diffusivity dominates and three sub-regimes in the Ohm-dominated
domain. Two of the Hall sub-regimes can be characterized as being ambipolar
diffusion-like and two as being Ohm-like. When the two-component plasma
consists instead of positively and negatively charged grains of equal mass, the
entire Hall domain and one of the Ohm sub-regimes disappear. In all viable
solutions the midplane neutral--ion momentum exchange time is shorter than the
local orbital time. Vertical magnetic squeezing always dominates over
gravitational tidal compression in this model. (Abridged)
