I'm not an expert here, hopefully somebody who really does know their stuff can answer this definitively. But anyway here goes ...
From http://www.physics-in-a-nutshell.com/article/4/lattice-basis-and-crystal "A lattice is in general defined as a discrete but infinite regular arrangement of points (lattice sites) in a vector space". Notice how this says nothing about atoms, molecules or ions; it is purely a regular layout of points in space. We define the atoms and their positions via the basis . This is simply a list of species types and their positions relative to an arbitrary origin. We then associate the basis with the lattice points, that is we use the lattice points as the origin for the basis, and as we have a crystal (and so translational symmetry) every lattice point is associated with the same basis. Thus to define a crystal structure
- We first define a lattice - a set of regularly laid out points
- We then define the basis - a list of atom types and positions and offsets
- We then associate an identical basis with every lattice point
Thus for a given lattice we can have any number of bases, and each basis can contain any number of atoms.
Now we can simplify things a little bit by noting that we can always shift the origin of the basis so that one of the atoms is at the origin, and thus in the crystal a lattice point coincides with one of the atoms in the basis. This is what has happened in your case c). Now note that for us here atoms are always point particles, so rather than large or small I'm going to use blue and red - I think giving the atom a size may be confusing here as here atoms only have a position and a type. So
- We have a simple cubic lattice, so the lattice points (which have nothing to do with atoms) are on a cubic layout throughout all space
- For the basis (i.e. the atoms) we have chosen to place a blue one at the origin, and a red one offset along the diagonal
- We then associate the basis with the lattice to make the crystal, the result being a blue atom at every lattice point (because the basis has a large atom at its origin), and a red atom offset diagonally from the lattice point, coincidentally in this case half way to the next lattice point due solely to the choice of basis, not the lattice
Also note that the basis may indeed be the positions and types of atoms in a molecule. We would then form a crystal of that molecule. But bonding is not important in this context, all we are interested in is the layout of the atoms in space, so for the moment we can forget all about chemical bonding and think purely in terms of atoms.