Everything I'm finding lists "Satomi" as just another Cornus kousa cultivar name and not as a species epithet.
Just looking at the different photos, I'm thinking it may be hard to figure out the cultivar definitively (unless you find a dogwood expert
.) 'Satomi' looks like a good candidate but, from looking at online photos, many of the pink kousas look pretty similar and flowers appear to be very mutable depending on the temperature and such.
Good luck!
Regardless of its ID, I've got to say that you got a very pretty gift!