difference between country and countryside

country - (Country is used to describe rural areas (our of town)?
He lives in the country.)
countryside - (Countryside is usually used when you are talking about the beauty or peacefulness of a country area)

country vs. countryside

(Is there a difference between these two words?

Countryside is used to describe the landscape?
The Cotswolds countryside is beautiful. ?

Country is used to describe rural areas (our of town)?
He lives in the country.

Would you say: This weekend I'm going to the coutry.
or This weekend I'm going to the countryside. ?

and

We have a holiday home in the country / countryside. ??)

• This weekend I am going to the country. Not countryside.
  
  We have a holiday home in the country. I think I have heard countryside before but it is very rare, the more common choice would be country.
  
  
  Your two definitions are correct. It is hard to state any rules when to use country and countryside, basiclally we just learn it by usage. But the two examples you gave are 100% correct.

“Go to the country”, “go to the countryside”

• Use country where the reference is geographical or to the fact that it is some distance from an urban area.
  Use countryside when you talk about the quality of the place, in contrast to the city.
  country (often the country) an area that is away from towns and cities, especially one with particular natural features: She lives in the country. ◇ an area of wooded country
  countryside land outside towns and cities, with fields, woods and farms. Countryside is usually used when you are talking about the beauty or peacefulness of a country area: a little village in the French countryside.

• Country or countryside both the words can be used for mentioning rural areas or areas that are not urbanized.