Ok, I'll bite.
tl;dr; Pack and cell are unambiguous, battery can mean either.
A factory that assembles cells (or modules) into packs is indeed a battery factory.
A factory that only combines cells or modules into packs is not a cell factory. This is a clear misnomer.
A factory that manufactures cells can technically also be called a battery factory due to single cell batteries being a thing. However, if the cells are never used singularly as built, then this is really stretching the term.
First off, the world lost precision when single cells were deemed batteries, but there we are. It's understandable why this happened since a device's singular battery can be discrete cells (AAA for instance), and one typically says "This needs new batteries." Where it really needs "a new battery" or "it needs new cells". However, the ambiguity leads to this 'what type of factory' situation.
A battery is a collection of one or more cells.
AA, AAA, C, D, etc are single cell, as are coin cell batteries (like in hearing aids or Tesla fobs.
A 9 Volt battery is 6 cells in one package.
A 12 V standard car battery is also 6 cells (though a different chemistry).
18650, 2170, 4680 are cell sizes. If a product only uses one, that is a single cell battery.
Some LFP are sourced as multicell battery modules themselves, not discrete cells.
EVs use packs made of multiple cells, thus packs are only multicell batteries, not single cells.
So I have it upside down and the media are correct.
Tesla is working to make cells and batteries. Ford/ GM making batteries.
Thanks I'll try to be better.