Does this have the same twin meanings as in English, then? That is, does it also mean 'country' as in 'countryside'?
Yes it does and you can also say "cefn gwlad" for this.
Ah, excellent! Thank you.
So you're saying it means rural region as well as a nation?
That seems quite colloquial
As mentioned above, you can use cefn gwlad for countryside too if you want to be specific.