It’s both.

Ok I checked the terminology and you're right, in Italian we call it "C natural" to distinguish it from the chord. Apparently in English you don't.

Context makes it clear whether the topic of discussion is the key of the song or a type of chord.

Also, in this way we can say "C major" or "C minor" to refer to the key of the song. We use "natural" when its important to distinguish from a note that has a sharp or flat - e.g. C# versus C natural.

In Italian, how would you refer to the key of Db major - "Db natural"? How would you refer to the key of Db minor?

You just specify the "minor", "major" is implied otherwise.