Why do hyperbolic projection models use a two-sheeted hyperboloid?

Question

I was under the impression that the reason hyperbolic geometry counters spherical geometry is that a hyperboloid has negative Gaussian curvature everywhere. However, it seems that this only applies to a one-sheeted hyperboloid, whereas a two-sheeted hyperboloid has elliptical geometry and has positive curvature everywhere. Why then, do models like the Poincare model and the Klein model utilize the two-sheeted hyperboloid?