In the US there are RNAV (GPS) and RNAV (RNP) approaches. The only RNAV (RNP) approaches I can find are at DCA. As you say, RNP requires not only that the box meet the standard but when it doesn't it tells you.
As I understand it, if your RNAV box is approved for approaches you can shoot RNAV (GPS) approaches. To do RNAV (RNP) requires another level of certification -AR. AC 90-101a and 105 give guidance.