I decided add this answer because I have a different viewpoint, although I agree with Wildcat on the IUPAC definition.
What exactly is an orbital? Atomic or molecular.
The term orbital is used with more than one meaning. Normally they are classified into different kinds. In a technical sense, they are just functions of the variables on only one electron. If the variables are spatial coordinates they are called spatial orbital, if they also include the spin coordinate they are called spin orbitals.
Generally speaking (for more than one electron particles) they are just mathematical constructions useful get the wavefunction or an approximation to it (from which we can get estimation of measurable values).
Is it the Schroedinger's equation solution, e.g., for Hydrogen atom?
I assume that you are speaking about the Time Independent Schrödinger Equation (TISE). In such case, generally not. In the particular case of H atom they are solutions of TISE, as well as linear combinations of them. In such case they represent stationary states of the system, in other words, are functions that represent (somehow) time independent states in which the system can be in (but only in systems with one electron!).
In polyelectronic species, they represent nothing, although it can be given (in some context) some approximate interpretations. Going further, if you decompose the system into one particle systems, and get their orbitals, it is mathematically true (because this: see last example) that they can be combined (how to is the hard part) to represent the wavefunction of the original system.
Is it the probability of finding any electron of an atom in any specific region around the atom's nucleus?
No. They are rather arbitrary. It is a common practice to transform them according to one needs. In any case, normally they represent something like the stationary state the one electron embed in a space region where it feels the mean interaction with the remaining electrons. But do not dig to much in those interpretations.
Maybe all these describe the same thing. What is it?
The introductory textbooks call orbital to an space region. It is not compatible with the usage of the word above. In a polyelectronic system, sometimes, for some kind of orbitals, in very well defined contexts, they can have a sort of interpretation, in essence they are just helpful functions.
You should check the answers to this question: What does an orbital mean in atoms with multiple electrons?... They give some details that I omitted.