Still, isn't the head of government on the de facto seat of power? The head of state, IIRC, is just ceremonial.
True enough. Which is why I said it was just a nit.
The President in most republics fills the ceremonial roles of the old absolute monarchs, and in theory appoints the premier and has the usually theoretical power to remove him from office.
The theory behind this is that the president is above politics, just as the British or Japanese monarch is, and thus can serve as a symbol of national unity without partisan issues getting involved. Parties and administrations come and go, but the nation endures.