From my point of view, an ab initio method is a method that allows you to predict the properties of a system from the first principles of the theoretical framework that you are using to describe your system (i. e. the only assumption is that the principles of your theory are correct). With this definition one can consider that quantum and classical mechanics, both, are ab initio theories in some sense. You have a set of first principles (quantum mechanics’ postulates or Newton’s laws) and you apply them to your system to obtain a set of properties that should be correct as long as the conditions of the system lie within the range of applicability of your theory.
Maybe an alternative (or complementary) definition of an ab initio method is that is a method that requires no parameters coming from experimental measures but, in my opinion it is difficult to put a threshold on that statement because any theory requires some fundamental constant that is not predicted by the theory (for example Newton’s gravitational constant, Dielectrical constant, fine structure constant and so on).
These are my thoughts on this topic.