Yes, the student gets a card that's clearly labelled with the name of the course:
The value of the card is questionable - very few operators outside of New Zealand have heard of my self reliant diver course, so it might as well say Golfball Retrieval Diver on the card......!
Distinctives are unique to the person who writes them and in order to be able to teach them then you need to submit a course outline to PADI - depending on the course, this could be a few pages or it could be 30+. Both my distinctive specialties are very borderline in terms of being "acceptable" to PADI - self reliant, for example, could easily be interpreted as "solo", so the outline also includes sample classroom presentations etc, with notes that indicate the correct emphasis for the course. it is a lot of work, it took me over a year to get it right - but in terms of having the freedom to teach what you think is right makes it worthwhile.
I believe that both SDI and TDI also have the option to write similar instructor-authored outlines?