Is it right if I say: Example: China
- Full Name: People's Republic of China
- Name: China
Example: United States
- Full Name: United States of America
- Name: United States
If wrong, then when to include the republican state's title?
Countries generally have official long and short versions of their name. This has nothing to do with whether or not they are a republic: for example, the Kingdom of the Netherlands and the United Kingdom of Great Britain and Northern Ireland. Normally, people use the short name but you can always use the long name if it makes things clearer.
In fact, sometimes, people use unofficial long names for countries for clarity. For example, Ireland is both an island and a country whose territory is about 85% of the island. Officially, the country is just called "Ireland", with no long form but people use the unofficial name "Republic of Ireland" to disambiguate between the country and the Island of Ireland.
In most cases, the short name of a country suffices. The long name is usually unnecessarily formal. However, you should use the official name in some diplomatically sensitive situations. Examples include:
You include the full name of a state when you don't want ambiguity about what nation you are meaning to identify. In English, "China" might refer to either the Republic of China [Taiwan] or the People's Republic of China. America might refer to the United States of America, or one of the continents in the Western Hemisphere. The "United States" could refer to the "United States of Mexico", the formal name of the country, or the "United States of America".
In most cases, especially informally, it is probably going to be acceptable to use merely "China", or "United States", or for the latter, even "USA".