There are white and green Brinjals too. Are the purple ones known by some special name?
In Britain (and France), the large purple varieties are known as aubergines. Other (pale and/or small) varieties aren't usually found outside of Asian supermarkets, where I imagine they are still referred to as brinjal.
The name 'eggplant' is used in the US, Canada, and the Antipodes, mainly because the lighter varieties are more common there, which arguably have the colour of eggshell.
They are, however, all just different varieties of the same plant, like yellow and red tomatoes.
They are all known as eggplant.
In the UK they are all known as Aubergines and in the US as Eggplant as they are all fruiting bodies of the same family of plants, the nightshades. The ones that are in common use in Europe tend to be the larger purple and white varieties as these are the ones that have been cultivated in that continent. With the spread of the cuisines of different countries into other parts of the globe, ingredients used in these cuisines have recently become more readily available outside their native countries and are called by names that distinguish them from their native equivalents. Personally, I buy what I know as Indian Aubergines and Thai Aubergines in my local Asian supermarket, these are imported from these countries and are not grown locally. I am grateful to know that the ones that are imported from Kashmir are known in that country as Brinjal. I would add that they all have a particular flavour/texture and are all of particular use in the cuisines of their respective native countries, after all, you wouldn't make a Moussaka with Brinjal.
In Toronto, we call everything in that family "Eggplant" (or Aubergines if you're trying to look cultured.)
The most common variety is the the large dark purple/black one. The second variety is the long skinny purple one, which we call "Chinese Eggplant".
Then there are the small ones, about the size of a pear or even smaller, which we call "Baby Eggplant" the default being the dark purple/black colour.
These distinctions really based on size and shape. Any further differentiation is done by pre-pending an adjective. "I'd like a light purple baby eggplant.