Is the lattice point in ZnO crystal structure occupied by the molecule ZnO or the ion O^2-/Zn^2+?
You are falling in the rather common mistake of confusing the crystal structure with the lattice. The lattice is a mathematical object (it does not exist as a real object), whereas the crystal structure is the convolution of the lattice with a basis (an atom or a collection of atoms).
In some cases atoms are distributed exactly at the same positions of the lattice points; for example Cu has a face centered cubic structure where copper atoms occupy the same positions of the face centered cubic lattice. In this case Cu at 0,0,0 position is your basis.
In your case neither the Zn atoms, nor O atoms occupy lattice positions. Your basis is contituted of Zn at 1/3,2/3,0 plus O at 1/3,2/3,z; 'attaching' this basis to each lattice point (taking into account the correct distances of both Zn and O atoms from each lattice point), you obtain the final structure.
In any case, the picture that you posted does not represent the real hexagonal structure; rather, I would that is the trigonal polymorph.