What are purple small Brinjals called in English?
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.