Did the British construe India as we know it?

Within Śrī Vaiṣṇavism coexisted two different interpretations of the way one's soul should surrender to God, one upholding the "monkey-way" (Vaṭakalai) and the other one the "kitten-way" (Teṅkalai). In both cases, the cub (i.e., the human soul) can do nothing on its own and completely depends on its parent (God). But whereas the baby monkey will at least hold on its mother's back, the kitten will be brought by its mother who seizes it by the scruff of the neck.
These two currents have enjoyed a certain fame even among Western scholars, due to the book Rudolf Otto dedicated to Viṣṇuism in 1917. There, he compares its split into two currents to the Western schism in the Christian Church (Kirchentrennung) and described the Vaṭakalai as the Roman Catholic option, with a Pelagian stress on human beings as able to attain salvation through their efforts and the Teṅkalai as the Lutheran option. (For a critique of the reception of Otto's sketchy description, see Hardy's article on the JIPh 1979:280).
By and large, Tamil authors tend to favour the kitten way and Sanskrit authors the monkey way. Vedānta Deśika is traditionally considered as the champion of the monkey-way. Or, at least, this is what I thought until recently. Srilata Raman (2007) notes that

it is at a very late date that the theological differences between the two schools of Śrīvaiṣṇavism come to be listed and formalised […]. It was only as late as the nineteenth century, the period when formal litigation in British courts began, that both the Vaṭakalais and the Teṅkalais needed to profile themselves as distinct separate sects, with irreconcilable theological differences. The formalization of hitherto fluid theological opinions in turn would have further helped consolidate sectarian identity (Raman 2007: 9-10).

As in other cases (the practice of satī, perhaps, the supremacy of Advaita Vedānta, the idea that the Vedas are the basis of all current practices, the concept of "philosophy", etc.), one notices that the British are not just a recent intervention in South Asia. They are now part of South Asian history and one cannot avoid them in one's hermeneutical enterprise.

Are readers aware of further cases of a "pizza effect", that is categories influenced by the West and then superimposed to older ones in South Asia and treated as "indigeneous"?

On Vedānta Deśika (deemed to be the champion of the monkey way), see here.