1) Country as a state/nation (country - the geographical area), is referred to as the area or land that has its own laws and government. There are some collocations that may help you to identify it in this usage (here some follow some examples ):
(a) foreign country;
a great major/leading/industrial country;
to rule/govern a country;
from different parts of the country (or several cities of the country...);
an oil-producing country;
an allied/enemy country;
...relations between the two countries...
and so on and on...a list would go forever, it would take all night and we wouldn´t exhaust all possibilities.
Country as an area away from towns and cities, with fields, woods and farms, is commonly used in descriptions of sceneries, particular features seen in travels, and the landscape in general. Hence, the focus would change from
the whole, or from politics and the macro-talking to a micro one . It would be used more adjectives as well.
That said, some of the collocations you could (much likely) come across in a piece of text/news etc.
-moutain/montainous/wild/rugged countryside (when talking about the local landscape, a region, etc, not the country as a whole);
-open country (here in the sense of scenery/landscape);
Countryside is usually used when you talk about the beauty and/or the peacefulness of a country/area.
Source:Oxford Learner´s Thesaurus -a Dictionary of synonyms (12th-Edition in print, not online.), mainly the collocation examples.
Ah, some of those collocations could be used to talk about one or the other meaning of the word, context is the name of the game.