We have constructed three-dimensional smoothed particle hydrodynamics models to examine the influence of a detached binary companion on the dusty winds from red giants and asymptotic giant branch (AGB) stars and the degree to which this model can reproduce some of the observable characteristics of axisymmetric or bipolar pre-planetary nebulae. In this second paper in the series, we focus our attention on the morphology of the circumstellar envelopes. The parameter space of our models includes wind outflow velocities in the range 10-26 km s-1, circular orbits with binary separations 3.6-50 AU, and binary companions having masses in the range 0.25-2 M☉. By varying these parameters, we find a continuous range of envelope geometries and density contrasts that correspond well to observational classifications of planetary and pre-planetary nebulae: bipolar, with density contrasts from 10 to >200 between the equatorial plane and the polar direction; elliptical, with intermediate contrasts of 5-10; and quasi-spherical, with very low density contrasts. This last category manifests a hitherto unknown type of hydrodynamic solution, in which a spiral shock is formed, covering most of the solid angle around the binary. The cross sections of these shocks, and to a lesser extent the two-dimensional projections of the quasi-spherical envelopes, appear as a series of rings in the wind. We discuss the observational implications of that type of wind solution. The quasi-spherical geometry is the prevailing type in parameter space. From binary statistics we estimate that ~34%-40% of detached binaries will give rise to bipolars for a 10 km s-1 outflow. We present a classification scheme of the envelope geometries based on a combination of binary and wind parameters. We also find that the mass accretion rates onto the secondary are systematically lower than is predicted by the Bondi-Hoyle theory.