That’s correct - formally known as Private Limited Companies.
It wouldn’t be unheard of a company in Scotland to have ‘Inc’ as part of their name. But it would only be part of the name, not as any legal title. ‘Inc’ signifies nothing about the legal status of a company based in Scotland.
Generally it would be used just 'cos it sounds impressive, and ‘big’ in an ironic multi-national American kind of way. You wouldn’t use it if you were really multi-national and big.
So you could even have a company called “Widgets Inc. plc.”
A name (including trading names) for any limited company is a legal title. However, there’d be no contradiction in having ‘Inc.’ within the title, because by definition all limited companies are incorporated. However, I’d suspect it would still get rejected by Companies House as potentially misleading.
Any use of ‘Inc.’ within a business name of a non-incorporated organisation would, I suspect, be running the risk of some serious questions being asked about whether it was potentially misleading.
The complexities of choosing a business name can be found here: http://www.companieshouse.gov.uk/about/gbhtml/gbf2.shtml
Clothes Inc Limited - UK Registered Company No. 04983581
Actually I’m not sure if that link takes you to the right place, but suffice it to say that Companies House has at least one UK registered company with ‘Inc’ in the name - Clothes Inc Ltd
Thanks all posters for an interesting discussion, and thanks **GorillaMan ** for pointing me to Stirling University.