It's familiar to me that the pseudosphere has local hyperbolic geometry (locally is isomorphic to a hyperbolic plane).
I am not very sure what is precisely presented in these pictures.
Also, I am interested in some examples of how the pseudosphere is related to the hyperbolic plane.
What I have is that the hyperbolic plane and pseudosphere have the same Gaussian curvature ($K = -1$), so they are locally isometric.
Any comments will be welcome.
You are correct about the local isometry.
The other three models show ways of mapping the hyperbolic plane bijectively, but not isometrically, onto an open subset of the Euclidean plane. They are analogous to map projections. In both cases, it is impossible to map the source (hyperbolic plane, or portion of the globe) isometrically onto the target because the curvatures are different, and curvature is an isometric invariant.
To help you compare the different models, they have picked one to draw a grid on, and shown what corresponds to it in the other models. In other words, they have mapped the grid to the hyperbolic plane, and mapped it back down to another model. They have also depicted a couple of geodesics (the green and blue curves). The vertical lines in the upper half plane are also geodesics. (I would consider them analogous to meridians in, say, the Mercator projection: these are also geodesic, and are depicted as vertical lines.)
The grid does not consist of equal sized squares according to the hyperbolic metric. It’s just there as a visual aid.
The mapping to the pseudosphere is a locally isometry. The drawbacks are (1) the target is a surface in 3D, instead of being a subset of the Euclidean plane, (2) the mapping cannot be continued past the rim, even though there is more hyperbolic plane “past it”, and (3) the topology is wrong: the hyperbolic plane is simply connected, but the pseudosphere has nontrivial loops.