I have found it next to impossible to find a definition of the hyperbolic plane. I am aware of all sorts of imperfect models of the hyperbolic plane -- the pseudosphere, the Poincare disc.
I also know several of its characteristics -- multiple "straight" lines through a given point not meeting another straight line not containing the point -- a surface in space of everywhere negative curvature satisfies the above hyperbolic axiom, etc. etc.
I would like a definition of just what the pseudosphere and the Poincare disc are imperfect models of. Wiki, Needham, Climenhaga etc. etc. do not provide explicit definitions. This is very unusual for a mathematical object of such interest.