While the official use of the term "company" may be regulated by their registration under the Companies Act in BE, the definitions for company from Collins and Oxford don't make that distinction:
Collins: 4. a business enterprise
Oxford: 1: A commercial business: example: a shipping company
Synonyms: firm, business, corporation, house, establishment, agency, office, bureau, institution, organization, operation, concern, enterprise, venture, undertaking, practice;
conglomerate, consortium, syndicate, group, chain, combine, multiple, multinational
informal outfit, set-up
Based on that, there is no basis to make the generalizations in #7 and #8.
There may be a technical difference between "company" and "firm" (even in AE, wow, imagine that!), but in "normal" use they remain synonyms (and, as Oxford claims, even in BE.)