The concept under description is effectively Bagehot's "dignified" vs. "efficient" parts of government. The head of state being the "dignified", the personification of the nation, and the head of government being the "efficient", the wielder of political power.

I generally favor the American-style presidential system over the British-style parliamentary system, but I think one flaw of the American one is that it combines the roles of the dignified and the efficient, often conflicting with each other, into one office.

The French semi-presidential system is a little weird, with technically the prime minister being the head of government, but the president still wields most of the executive power.