Is that common in the US? Because here in Sweden it is regarded as bad practice for medium sized companies and up. The CEO is almost never part of the board here except for in tiny companies.
In fact, I have seen this very debate play out in real life. We appointed a new President of the ASF during my term. That person had been a Director, but when they became President they stepped down from the Board.
For a non-profit charity that relies on volunteers, perfect separation is not always feasible, and especially in the early years when the ASF was smaller the President was often a Board member. But although I can't speak for anybody but myself, I believe that the ASF is likely to continue with an informal tradition of separating Board from Officers for the indefinite future.
A similar cultural change ought to be possible at Mozilla — and perhaps elsewhere.
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.