Pakistan's constitutional setup is somewhat interesting. Executive power lies with a prime minister who is responsible to a parliament, but the country also boasts a powerful presidency. The president can sack the prime minister and call elections, and is also head of the country's armed forces. Crucially, perhaps, it is not the prime minister but the president who controls Pakistan's nuclear arsenal.
The Pakistani president is not directly elected. Rather, it is the country's parliament who elect the president. This makes Pakistan a bit of a taxonomical anomaly. It is not semi-presidential, because every definition of semi-presidentialism requires the president to be popularly elected. But it seems problematic to think of the country as a parliamentary system, given the power over parliament exercised by the president*. And while the powers of Pakistan's appointed president are perhaps unusually high, the country is not unique in having an unelected president who is a serious player in the country's politics. Off the top of my head, the Czech Republic, Israel, and (to an extent) Italy spring to mind as countries where the president is appointed by parliament but plays more than a purely ceremonial part in national politics.
Definitions of semi-presidentialism focus on the direct election of the president**. This leads to the lumping together of countries with powerful activist presidents and ones where the president has a purely symbolic role. This is not necessarily that problematic, as you can then go on to ask interesting questions about why one directly elected president is powerless while another is the centre of their country's politics. But maybe in another way it misses something. Power is surely the currency of politics, and what makes semi-presidential systems interesting is the (real or potential) presence of two loci of power. By focusing on how presidents are appointed, semi-presidentialists look at one type of dual-executive set-up but ignore others. This does seem problematic, as you can end up analytically separating political systems that end up closely resembling each other.
If I was in the business of further academic research, one thing I would consider looking at would be some kind of comparison of presidents in countries where they are elected and countries where they are appointed by parliament. One thing I have picked up is that there has been relatively little academic research on appointed presidencies, even where these are players in their countries' politics, and this strikes me as an obvious gap that needs filling.
*this is aside from Pakistan's status as a country of questionable democratic credentials, where real power is exercised by a variety of entrenched yet dysfunctional elites immune from electoral accountability
**and their being faced by a prime minister responsible to parliament, obv.