This is another post about semi-presidentialism, triggered by a comment to my last post on Timor-Leste. I understand that many people are not so fascinated by semi-presidentialism that they want to hear more about it, but I am a slave to those who pose me questions.
A mysterious commenter identifying him or herself only as Rhunzzz mentioned the concepts of premier-presidentialism and president-parliamentarism, originally formulated by scholars Shugart & Carey. My recollection is that they took a dislike to the term semi-presidentialism and attempted to coin new terms they felt were more descriptive. Their terms seem to have caught on to some extent, but mainly as explicit subtypes of semi-presidential regimes. The premier-presidential type sees more power in the relationship going to the prime minister, but with the elected president still doing more than nothing, while their definition of a president-parliamentary regime is one where the president can sack the prime minister or individual ministers (but I think with their replacements still being subject to parliamentary approval).
As is the way of the human world, it is difficult to precisely assign semi-presidential countries into one or other regime type. French presidents, say, often act as though they can sack ministers and prime ministers, but they do not actually have the constitutional power to do this; they can force ministerial resignations if they are the head of the government party or coalition, but not if the Assembly's majority is united against them. For all that people do use the premier-presidential v. presidential-parliamentary distinction, it all seesm a bit hair-splitty in practice.
In the Palestinian Authority, the president can sack a prime minister, but the sacked prime minister remains in office in a caretaker capacity with their government until the Palestinian parliament chooses a successor. It is not obvious to me that the president can sack individual ministers, which reminds me that I need to download the Basic Law of the Palestinian Authority. I think that it the president can only sack the prime minister but not individual ministers then we are in premier-presidential territory. For more on the prerogatives of PA presidents and prime ministers, I refer you to Nathan J. Brown's "What can Abu Mazin do?".
In Timor-Leste, the constitutional powers of the president are pretty limited, which is probably one reason why Xanana Gusmao has chosen to move from the presidency to the premiership. Gusmao as president was nevertheless able to use his personal prestige and massive electoral mandate to assume a more than ceremonial role in the country's politics, playing a part in forcing Prime Minister Alkatiri from office.