It's hard to use them consistently because there isn't a single universally accepted definition.

Most people would consider the Netherlands a "country", but now we have a country within that country. Israel is a state, Japan is a state, but there are 50 states in the United States. "[People's] Republic of XYZ" generally refers to a sovereign state, but Russia has republics inside. You can't just call something what the locals call it and expect that your readers will get the picture. Even worse, people are often deeply divided as to what a given territory should be called.

So I will generally forgive journalists for picking a neutral-sounding, ambiguous expression in cases like this. What matters here is that the Dutch control this airspace, regardless of Curaçao's status within their kingdom.