But how do you define 'culture, language and the art of living', as practiced by the people in the regions you mentioned, to be an 'exclusive continuation' of the 'culture, language and art of living' practiced by the IVC?
That's a wrong way of seeing it. You're comparing between two extreme end of a long and tangled thread and deciding that the thread is snapped somewhere as the two ends don't meet.
Consider it as a long line where, to name a few - IVC, Cemetery H culture, Gandhara Grave Culture, Early Vedic period, Late Vedic period, Mahajapadas, Mouryas, Guptas, Palas, Sultanate, Mughals, British India, Republic of Indian/Islamic Republic of Pakistan etc are points which make the line. Now the correct way of judging continuity would be to compare each adjacent points and see if there is any sudden break in Civilizational Continuity.
Again, does a shift to agnosticism/atheism, English as a primary language, Western dress and customs, signify a dilution of the aforementioned 'culture, language and art of living'?
Shifting to agnosticism/atheism might be considered break in continuity in case of Abrahamic religions, but Indian religions haven't been organized as such and atheism/agnosticism have always been part of Indian philosophy. You might as well look up Cārvāka and other materialistic movements such as Buddhism.
English never is a primary language. It's the language of the court, just as once Sanskrit was while the mass spoke Prakrit(another example is Persian -- Hindustani). There was time when British royalty preferred to speak French, did that break the continuity of Britain's history?
Will the residents of the regions you mentioned soon be considered 'a continuation of Western Civilization' (instead of the Vedic Civilization) because they are 'agnostic/atheists, get drunk at pubs, eat meat or whatever the feel like, practice liberal social values'?
I have replied to most of the above points. They will only be considered as added values. Just as foreign loan words enhance the indigenous language, but don't make it alien.
That said, why would we want to claim that any particular civilization has ownership of values (liberal social values) that appear to be common sense to many people?
That I never claimed, all civilizations have been liberal at some point of time and sometimes they shunned all changes.