Is two-dimensional hyperbolic geometry unique up to isomorphism? This may have been answered here somewhere, but I'm unable to find it.  
Is 2D hyperbolic geometry is unique up to isomorphism (or up to whatever's appropriate)?  I know there are at least four models -- Poincare disk, half-plane, etc., but I assume they're essentially the same in some sense?
Another way to put my question:  suppose we keep Euclid's first four postulates, but replace Playfair's equivalent of the parallel posutulate ("Given a line  L  and a point  P  not on  L, there exists at most one line through  P  that doesn't intersect  L)  by, "Given a line  L  and a point  P  not on  L, there exist at least two lines through  P  that don't intersect  L."  Can we deduce from the five resulting axioms that at most one geometry that results?
Thanks for any help you can offer!  References would especially be appreciated.  I'm an analyst, so I know how to read math, but I'm not an expert in this area, so I'm looking for sources that are accessible to mathematicians generally.
Best Regards,
Bob
 A: The appropriate notion here is isometry: if points $a$ and $b$ are mapped to $a'$ and $b'$ respectively, the distance between points $d(a,b) = d'(a',b')$. $d$ and $d'$ here denote the represented distance, not the Euclidean distance between the points in the model. Geometry is about measuring distances, so to say.
Assuming four postulates of Euclid plus your variant of the Playfair's axiom, you have that the curvature is fixed and negative, but the specific value of curvature is not known. With curvature -1, a triangle with four angles 45 degrees each will have an area of $\pi/4$. If the curvature is, say, -2, this triangle will have an area of $\pi/8$; in general, the sum of angles of a triangle minus $\pi$ equals area times the curvature (or, for surfaces where the curvature is not constant, the integral of the curvature). The curvature is only the matter of scale (a larger sphere will have smaller (positive) curvature, but it is essentially the same shape).
When we fix curvature -1, all the common models (Poincaré, Klein, hyperboloid, half-plane) are isometric (my page lists the common models and several less common ones; but I have no references for how the postulates plus curvature fix the geometry). An useful analogy: cartographers use many projections of the surface of the sphere (stereographic, Mercator, etc.) but they all describe the same mathematical object on a flat 2D map, and since the sphere is not flat, none of them is perfect, and they have different advantages and disadvantages. The same is true about the models of hyperbolic geometry.
Surfaces of constant curvature need not be isometric to hyperbolic plane, because they can correspond to only a fragment of a plane (a disk is not isometric to the whole plane, even though both have curvature 0) or they can be wrapped (a cylinder is not isometric to the whole plane, even though both have curvature 0 -- this happens with the tractricoid aka pseudosphere, which is listed sometimes as a model of hyperbolic geometry). However, such cases do not satisfy the postulates.
The isometric mappings between the common models not only exist, but they are also given by simple formulas. For example, the mapping between the half-plane and the Poincaré disk is inversion, and Klein/Poincaré/Gans models are obtained from the hyperboloid model with a simple perspective transformation.
