I'd tend to say to another family member that I met firstname secondname, because there are lots of duplicate first names (but often with clarifying abbreviations, so Willie, Bill, Liam and William are clearly themselves, despite all being Williams), but introduce them to a stranger simply as 'this is my cousin firstname'. If it seemed appropriate I might clarify further, but not necessarily.
I used to have 47 first cousins of my own generation, a few have dropped off their perches, but the others have produced an ever widening pool of 1st removeds and 2nd removeds, many of whom I don't know, but they're still my cousins, as are the seconds and thirds.