ftr said:
It seems that the word ontology which suppose to be about the most concrete object we can come up with is itself not well defined.
I would disagree.
I would accept that you reject Demystifier's notion of "primitive ontology" as not sufficiently well-defined. But the ontology is sufficiently well-defined. ("Sufficiently" means that I'm talking about notions of what is well-defined for physicists, not for mathematicians.) In particular, the ontology should contain all the information necessary and used to predict the evolution in future.
So, for some classical theory with Lagrange formalism, the ontology should contain ##q(t_0),\dot{q}(t_0)##, if it is given in Hamilton formalism, it should contain ##q(t_0), p(t_0)##. In dBB theory, it should contain ##q(t_0),\psi(q,t_0)##. This is not restricted to deterministic theories, say, for Brownian motion the ontology would be simply ##q(t_0)## and the predictions about the future would be only statistical.
Let's also not forget that the ontology of some particular system, even if it is handled like a closed system, may nonetheless depend on the external world. Thus, the complete ontology may contain the ontology of the external world too. So, in an ##\psi##-epistemic interpretation, ##\psi(q,t)## describes only our knowledge about the ontology of the system itself. But this knowledge may be objective - the measurement device used for preparation, the measurement result of the preparation measurement - and part of the ontology external to the system.
The precision of this definition is quite restricted - in principle, "theory with ontology" is a metatheoretical notion, which should be applicable to any theory. But the set containing "any theory" is not a well-defined set in mathematical set theory. All we can do is, therefore, to clarify the meaning of ontology for all theories which claim to have some ontology. This is usually not a problem.
If it is a problem, it is a strong indication that the theory is not well-defined. I see, for example, no definition of the ontology in many worlds which would make sense. There is a definition - all what exists is the wave function. But it makes no sense, because all the things they talk about, like the worlds themselves, are not well-defined by the wave function alone.