The band theory of solids has developed to describe the collective behavior of atoms and molecules when in the solid form.
The confusion goes away if one realizes that different quantum mechanical models have developed in order to describe different problems.
For an atom, can having at least one electron in the conduction band mean that the atom has been ionised?
In order for an electron to be considered in the conduction band, one needs a solid. Individual atoms/molecules sit at the lattice sites of the solid. The model is that the conduction band has electrons that are bound over the whole lattice. The valence band electrons are tied on the points of the lattice where the nucleus also sits (depending on the material, whether insulator or conductor).
In the sense of the band theory model, the whole lattice is still neutral. Just the conduction band electrons have definition over the whole lattice ( a quantum mechanical probability to be anywhere in the lattice). It has little meaning to talk about an ionized atom, because its "own electron might be away but another in the conduction band is near", (as a hand-waving explanation).
Can having an electron in the donor energy level or acceptor energy level mean that the atom has been ionised?
If for individual atoms, yes, but if the atom is part of a lattice see above.
Is free electron same as an electron in the conduction band or an electron in the acceptor or donor band?
No a free electron means there is no potential that is binding it.
Is an atom having a free electron an ionized atom?
If the atom is not in a lattice and it attracts a floating electron, yes it will be a negatively ionize atom. If in a lattice,see above
Maybe if you read the whole link your confusion will go away.