This is a long reply to a question posed by Nicholas Whyte in a comment to my last post. He asked whether Cyprus and former Soviet states count as semi-presidential or not. This leads me into a discussion of what constitutes a semi-presidential regime. Unfortunately, there are several definitions of semi-presidentialism, so the answer has to be "it depends". The first English language definition by French scholar Maurice Duverger talked of countries being semi-presidential if they had a popularly elected president who had "considerable powers" but how faced a prime minister who led a government that was responsible to an elected assembly. The "considerable powers" business then leads to considerable debate as to whether a given president in a given country has considerable powers or not. It can also be difficult in practice to identify what powers a president actually has, given the divergence that can occur between a president's constitutionally granted powers and the powers they wield in practice (compare the limited powers of the French president in the constitution with the powers they have actually exercised). Nevertheless, despite these problems, Duverger's definition and definitions derived from it are probably still dominant.
Robert Elgie attempted to produce a semi-presidentialism definition that allows for a more precise determination of whether a country is semi-presidential or not. He skips all that considerable powers stuff by saying that a regime is semi-presidential if it has a popularly elected president and a prime minister responsible to parliament. Using this definition, a semi-presidential country can have a very powerful president, or one who spends his or her time playing golf. Some find this kind of definition problematic, as it includes countries which in practice have politics so similar to parliamentary regimes as to make no difference; if you think the idea of bifurcated power structures is crucial to any discussion of semi-presidentialism as a regime type then Elgie's definition is not for you. However, even with that, a thing you do see in country's with figurehead directly elected presidents is that sometimes they can leverage their status as someone chosen by the people as a whole to exert pressure on the government in a manner bearing little relation to their paper powers. So even a figurehead president might be able to restrain a prime minister, in certain circumstances. Arguably, this kind of happened in Timor-Leste, where President Gusmao announced that he had lost confidence of Prime Minister Alkatiri, increasing the pressure on the latter that ultimately led to his resignation.
And so to the countries Nicholas Whyte mentions. I can't speak for Cyprus, as I don't know too much about how day-to-day politics works in either jurisdiction there. With the countries of the former Soviet Union, political systems there vary greatly. Many of the USSR's successor states are straightforwardly dictatorships. However, most of the ones that do still have some kind of democratic politics in place count as semi-presidential, with the regime-type being apparently more common in former communist countries than either straight presidential or parliamentary regimes. Or so I have read. I have my doubts with some of the countries. Take Russia – it has the prime minister and president you associate with semi-presidentialism, but the prime minister is only responsible to parliament in the most notional of manners. In practice (and probably in the constitution as well, given that it was written by a president who then had it passed at gun point) the prime minister is the president's bitch, someone he gets to look after tawdry day-to-day stuff for him.
Ukraine is more like a classic semi-presidential regime, where the prime minister and president are both powerful, but the prime minister is genuinely responsible to parliament; I think the relationship betweern the offices has changed over time, and the presidency lost a lot of its powers recently as part of the deal that saw Viktor Yushchenko elected to it. I gather that Lithuania is more the kind of country that scrapes into the semi-presidential category only if you use the Elgie definition, as its president is directly elected but aloof from actual politics. I think the rest of the democratic ex-USSR states are at least Elgie semi-presidential, except for Latvia and Estonia, which I believe to have adopted parliamentary regimes at independence, and Moldova, whose parliament changed it to a fully parliamentary regime, much to the chagrin of the then president who wanted things moved in a more fully presidential direction.