Indian here. Extremely common for the CEO to be part of the board. In fact the CEO equivalent "Executive head of company" designation in India before globalization made use of CEO fashionable was 'Managing Director'.
The legal world has "managing partner" as the equivalent for LLPs.
It can vary, though; e.g. Kevin Sneader is the de facto CEO of McKinsey, but his official title is Global Managing Partner and his predecessor used Managing Director.