Presidential v. Parliamentary Systems of Government
In parliamentary systems of government, the head of government can be removed any time that the prime minister or premier loses the confidence of a majority of members of parliament for any reason.
In strong Presidential systems of government (i.e. systems where the President has more policy making powers than those of a primarily symbolic constitutional monarch), including the United States, in contrast, (basically by definition) the head of government does not serve at the pleasure of the legislature, but may sometimes be removed by the legislature in extraordinary circumstances.
For these purposes, I would consider both "Presidential Republics" and "Semi-Presidential Republics" to be strong Presidential systems, but not "Parliamentary Republics", with or without a ceremonial/non-executive President. There are 41 Presidential Republics in the world in widely recognized countries other than the U.S., more than a third of which are in Latin America, and 23 Semi-Presidential Republics (not counting Iran which is closer to a constitutional monarchy).
Impeachment In The United States Is Only For Cause
In the United States, the process of removing a President is called "impeachment" and Article II, Section 4 of the United States Constitution provides that it is only permitted for cause, stating:
The President, Vice President and all civil officers of the United States, shall be removed from office on impeachment for, and conviction of, treason, bribery, or other high crimes and misdemeanors.
This is somewhat less restrictive than it seems, because the courts have determined that what constitutes "high crimes and misdemeanors" is a "political question" not subject to judicial review. See Nixon v. United States, 506 U.S. 224 (1993) (concerning a judge with the surname Nixon, not the former President of the United States).
But, a legislator is sworn to uphold the U.S. Constitution. See U.S. Constitution, Article VI (which provides in the pertinent part: "The Senators and Representatives before mentioned, and the members of the several state legislatures, and all executive and judicial officers, both of the United States and of the several states, shall be bound by oath or affirmation, to support this Constitution; but no religious test shall ever be required as a qualification to any office or public trust under the United States.")
So, in theory, a legislator applying the impeachment provisions of the U.S. Constitution has a moral duty, at least, to only remove a President from office via impeachment for conduct that amounts to some sort of crime, even if this duty is unenforceable in practice.
Are there countries with strong Presidential systems in which, unlike the United States, a supermajority can impeach a President (or undertake some equivalent process allowing a national legislature to remove a President from office mid-term) either without cause, or with a definition of good cause that is far more broadly defined than it is in Article II, Section 4 of the United States Constitution?
I'm looking for any examples of such constitutional or legal provisions, not necessarily a comprehensive list of them.