The term presidential system in Political Science isn’t really that ambiguous: one typically talks about democracies falling into Presidential, Semi-Presidential, and Parliamentary systems.

One thing to realize is that a democracy can have both an administratively powerful Head of Government and a largely symbolic Head of State. In America, these two roles are combined in the same job: President. In the UK, and other constitutional monarchies, the Queen (or King) is the largely symbolic Head of State with very few actual powers and the Prime Minister is the actual power (of course, power shifted from the former to the latter over the course of centuries). Confusingly, in some Parliamentary systems, the Head of Government is the Prime Minister but the Head of State is called the President (Ireland and Israel both do that). The Semi-Presidential system (most famously France and its colonies, but also Poland, Russia, and Egypt while it was a democracy and several others) can also be confusing, because they have a parliament and a president, like Ireland and Israel, but in this case the president will hold real administrative day to day power. These systems differ but, to give you a sense from the French example, a very very rough outline is that when the President and Prime Minister are of opposing parties (this is generally called “cohabitation”), the Prime Minister is responsible for most domestic affairs and the President is in charge of most foreign affairs. The President, however, can dismiss the Prime Minister (a new one will be ultimately selected by parliament). When they’re of the same or coalition parties, the President has de facto control over all areas of policy and the Prime Minister functions as a cabinet minister.

Now why are pure presidential systems less stable? One reason is that it’s common for the executive and legislature to be of different parties. In fact, in America for example, this has long been the norm—full control is the exception. In some cases, they can cooperate “across the aisle” and bang out compromises, but in many cases you have a frustrated executive who the people treat as responsible for government but who can’t get their priorities through the legislature. The democracy ends in these cases in one of two ways: either the president seizes complete power (often claiming a popular mandate) or the military steps in to end this gridlock (often claiming too they have a popular mandate to “fix” the government). Many Semi-Presidential systems (again, the classic example is France) have altered election schedules so that they’re more aligned and you have less chance of cohabitation. In Presidential systems, however, it’s common for legislative elections and presidential elections to not fully be in sync. For example, in the US, Presidents are on a four-year cycle and the legislature is on a two-year cycle. In modern history, almost invariably, no matter how popular a president is when elected, they frequently will lose seats in Congress during “midterm” (ie in the middle of the President’s term) elections.

Classically in parliamentary systems, dissatisfaction with the leadership can lead to removal through votes of “no confidence” which can leader to either the just prime minister being replaced or even full new “snap” elections. There is a much quicker feedback loop to popular dissatisfaction (if it effervesces up to the ruling party/coalition, at least) than in presidential systems, where—barring impeachment—the president will serve their whole term no matter their unpopularity. During that term, they may move to take extrademocratic steps to preserve their power despite their unpopularity. The checks against this can be limited—in a parliamentary system, even ones own party can be a check (after all, many ambitious members of parliament hope that they, too, will be prime minister after the next leadership election).

In the presidential system, both the legislature and the president will claim to have the popular mandate and, when they clash and get mired in gridlock, the president may take over unconstitutional powers (they are commander-in-chief of the military) or, if the president and legislature blame each other for the gridlock and there’s some popular frustration, the military may step in to “end the gridlock”. In a parliamentary system, gridlock often leads to ne elections. Likewise, in a parliamentary system, typically the largely ceremonial head of state will have a few powers to defuse any crisis, often again by just calling new elections. Unpopular leadership can be changed in constitutional ways, whereas in presidential system unpopular leadership may end up seeming largely unaccountable. Especially in a crisis, there may be a sense that either the president or the military needs to “step in” to end certain constitutional protections (i.e suspend the legislature or remove the president, respectively) in order to deal with the crisis.