I used to manage lithium-ion business for a large Japanese manufacturer in North America...
Cell was the term we used for the bare cells without protection circuits or outside packaging. This is the "can" part; the bare component.
Battery was used for finished packs whether they are single cells or multiple cells together with protection circuits and outside packaging. Finshed product.
Primary cells are complete packages within themselves so they may be called batteries.
For example, 9V alkaline batteries (the square one) are "batteries" but if you take one apart (not recommended), you may find six AAAA cells in them. I call AAAA cells "cells" because they are bare components.
Michael