Yes as a result of the empire if a British person and British Asians says Asian they mean for the Indian sub continent or those Asians ethnically cleansed from Africa in the 60's and 70's
You do know that India is in Asia, right? Also, Sikhs more often than not are Indians. Sikhism just like many of the world's other religions (Hinduism, Buddhism, Jainism) was born in India.
For Americans, "Asian" usually means the people we fought with or against in the Pacific in WW2, Korea, or Vietnam.
For a significant fraction of Americans "Indian" evokes visions of cowboy antagonists or casinos, rather than the Taj Mahal. That's not politically correct, but that is what it is...