The last Mughal emperor might disagree.
"Indian sub-continent," "South Asia," "Indo-Pak," all attempts to categorize an area that looks like part of Asia from space, but culturally and historically doesn't fit as well as a square peg in a round hole. Much like Mexico resulting in the term "Latin America."
I would also say the reason British influence did not extend beyond port cities into the interior of China at the time of the Mughal Empire is because it could not, not because the British were content to limit themselves to India. British endeavors around the globe at that time offer ample support of my point. If there was any confusion as to my point, at any time in modern history at least one of the three countries I listed could have accomplished the task, not all at every time or every smaller nation in the region. Even in WWII it took an Asian power(China), a European power(Russia), an American power(US), and even a bit from Australia to push the Japanese back after they successfully removed all European and American influence with alarming speed and ease. Remember the Japanese, not the Germans, fielded the largest battleships in history. They also engineered a functioning bridge over the River Kwai without any help from British prisoners, who were used for thoughtless manual labor labor(read up on how ticked a lot of both British and Japanese involved in that project were over the movie).
Had China wanted to wipe the British out of the interior of India in 1850 I think they likely could have. The British would have shelled a dozen or so Chinese cities and had the effect of a WWII firebombing though. Historical guesses as to the population of those cities varies widely, but by all accounts they were densely populated, highly combustible, and housed a sizable population. Beijing probably being the only city in the world with over a million residents at the time. Along with the loss of life, the British would have almost certainly been able to blockade every major port and navigable river.