Rock salt structure: chloride lattice or sodium lattice?
Unit cells are just a way of describing a given crystal. You can certainly draw $\ce{NaCl}$ the other way, with $\ce{Na+}$ in the corners and it's still correct. There are a few ways of thinking about this structure:


as two interpenetrating FCC lattices
as a $\ce{Cl-}$ lattice with $\ce{Na+}$ in the octahedral holes
as a $\ce{Na+}$ lattice with $\ce{Cl-}$ in the octahedral holes


They're all equivalent. The figure above actually shows both types of octahedral holes. Some might prefer to visualize the structure as the second case since, as you say, the chloride ions are bigger, but going the other way isn't wrong, nor does it describe anything different.