Scott,
short answer is company_name will be changed for all contacts within this company in one case, when company_id also present in request and not changed and new company_name is unique.

Full answer with all cases for company change in contact update:

Full update (without partial param)

1. company_name is only provided
   • with multiple contacts within company, contact will be moved to other company (split)
   • if last in company, contact will be moved to other existing company and error will be raised if company has some details and cannot be removed
   • if last in company, company will be renamed if name is unique
2. company_id is only provided
   • with multiple contacts in company contact will be moved to other company
   • when last in company, contact will be moved to other company and error will be raised if company has some details and cannot be removed
   • Error will be raised if invalid ID will be provided
3. Both company_id and company_name are present
   • when company_name is only changed, contact will be moved to other company if company with given name exists
   • when company_name is only changed, company will be renamed if name is unique
   • will raise error if invalid ID will be provided
   • when last contact in company, move will be allowed only if company does not contain any data
   • when both values are different, contact will be moved to new company and the other company will be renamed; error will be raised when new name is not unique
   • if any of those will be set to null, contact will be removed from company
4. neither company_id nor company_name are present
   • when multiple contacts in company, contact will be removed from company
   • will remove company details only if it is not containing any data
   • company cannot be removed from contact which does not have lastname

Partial update (with partial=1)

1. company_name is only provided - the same behaviour as in case of full update
2. company_id is only provided
   • contact will be moved into new company
   • if last contact in company, error will be raised if company contains some data
   • error will be raised if invalid ID will be provided (i.e. company does not exists)
3. both company_name annd company_id are present - the same behaviour as in case of full update
4. neither company_id nor company_name are present
   • no action on company data will be taken

I hope this answer helps.

Update : corrected rules for case company_name is only provided.

Kind regards,
Mikhail