We consider geometries and physical models for weak low-ionization absorbers based on the relative incidence of low-and high-ionization systems. We present a survey of weak low-ionization systems in 35 high-resolution HST STIS quasar spectra, sometimes supplemented by Keck HIRES and HST FOS data. We found 16 metal-line systems, with low-and/or high-ionization absorption detected. Weak low-ionization absorbers trace an abundant population of metal-enriched regions. Models show that these systems have a $10 pc region of higher density gas and a $1 kpc region that represents a lower density, higher ionization phase. The goal of our survey was to compare systems detected in low-and/or high-ionization gas. All but 1 of the 10 weak low-ionization systems have a related highionization phase. In three cases the high-ionization gas has only a single component, kinematically centered on the low-ionization absorption, and the other six cases have additional high-ionization components offset in velocity. The high-ionization absorption in weak low-ionization systems has similar kinematic structure to that in high-ionizationonly systems. There are just six systems with only a high-ionization phase, as compared to the nine systems with both low-and high-ionization phases. We conclude that filamentary and sheetlike geometries are favored, due to the relatively small observed cross section of high-ionization-only systems. Our statistical arguments suggest that although low-ionization absorbers are not closely associated with luminous galaxies, they arise in their immediate environments within the cosmic web.