The World and the Public

I'm not quite sure I understand your examples, Guest-- are those four different examples? One explanation is that the adjectives would be lower-case ('world-wide', 'public-minded') and the nouns could be capitalized if the words represent the entity personified, as an agent, etc:

'The World/world is aghast at the atrocities in Trepsyvanica.'
'The Public/public will not stand for any more political prevarication.'

But it is not required.