Well, the Bombay Presidency draws its 2nd half of the name from the fact that provinces of India were at one point called "presidencies" under (early) East India Company rule.
During the period of Company rule in India, 1757–1858, the [East India] Company gradually acquired sovereignty over large parts of India, now called "Presidencies".
Under the British Raj (1858–1947), administrative boundaries were extended to include a few other British-administered regions, such as Upper Burma. Increasingly, however, the unwieldy presidencies were broken up into "Provinces".
By the mid-18th century, the three principal trading settlements including factories and forts, were then called the Madras Presidency (or the Presidency of Fort St. George), the Bombay Presidency, and the Bengal Presidency (or the Presidency of Fort William) — each administered by a Governor.
So it seems "presidency" was a larger "province" in mid-18th century British terminology, or at least in East India Company terminology. I'm not enough of a British history fiend to tell you why they preferred that "presidency" term back then for such a territorial division. Wikipedia also seems to suggest that a Governor[-in-Council] was assimilated to a "president", under that system:
The government of Bombay was administered by a Governor-in-Council, consisting of the Governor as president and two ordinary members.
So perhaps that's the origin of the presidency term in the sense that the British used it, i.e. that the Governor-in-Council was like a president (of an executive), but I'm not entirely certain. M-W defines
governor-in-council: the governor of a British colony acting with the advice and usually in the presence of the executive council but not always with its consent.