At least for this vegetable berenjena seems to be the universal word not like other vegetables. You definitely have to know the local name or you may not be understood or get something totally unexpected. When I visit different Latin countries I always enjoy going into the grocery store and browsing in the fruit and vegetable section just to see and learn the local names. The differences and variations are always interesting.
Let's not forget the expression that's a spin off from berenjena:
I had almost forgotten that phrase, Lesley. It is true, berenjena must be of one those few words for vegetables that is universal. The names of fruits and vegetables do change, even from one island to the other in the Spanish speaking Caribbean.