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'.
Yeah I remember that. While growing up in 90's and even early 2000's, I always heard praises of someone reaching post of MD. The term CEO became common later.
There are still plenty of MDs in the world, but these days it's mostly used in finance and consulting to represent the first layer of leadership below the c-suite.
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.