what is a transverse isotropic rock?
You are correct. Isotropy means that the property at some specified point (X,Y,Z) is the same, whether the measurement is made along any direction.

If the property varies depending on which direction you make the measurement in, then that property is said to be "anisotropic". For example, the permeability of a material (i.e. its ability to conduct fluids) may be anisotropic if fluid can move preferentially in one direction and it would be difficult for it to move in a different direction (like the tubes below). This means, the permeability of this material at a given point (X,Y,Z) depends on which direction you make the measurement, i.e. high permeability parallel to the tubes, low permeability otherwise.



Transverse isotropy (also called hexagonal symmetry) is a special kind of anisotropy in which there exists a special "axis of symmetry" such that the property you are measuring appears to be the same as you make measurements perpendicular to this axis. Basically, rotating around this axis of symmetry, the property looks pretty much the same, i.e. it is isotropic in this transverse direction. However, once you go off axis, then the measured property will change.



In geology (especially in sedimentary geology), transverse isotropy is usually a good approximation to many phenomena (elasticity, permeability, heat conductivity, etc.) due to the fact rocks are deposited mostly horizontally. This means that things will look very similar laterally (in both N-S and E-W directions) but very different vertically.



Finally, transverse isotropy is just an approximation to the much more complex nature of real rocks. If you were to add multiple sets of fractures to a rock on top of its layered nature, you will need to use a "less symmetric" model to describe it: orthorhombic, monoclinic, or triclinic. But for most practical purposes, transverse isotropy does a pretty good job.

Hope this helps!