Well I meant more established, stronger democracies. Egypt was a post revolutionary state whose democracy is quiet weak, same goes for those other nations listed. Not saying I’m all right, but it’s rare for established democracies and that is what I originally meant.
Interestingly, last year I started a thread asking about the second oldest presidential system, after the US, and the consensus here on the SDMB was Costa Rica: What is the second-oldest presidential/congressional republic?
It does seem a pretty strong empirical measure of the presidential system that there is only one example of a country which uses it that has lasted for more than a century.

The situation in Portugal, a parliamentary democracy with a prime minister at the helm and ceremonial president as head of state, also shows some of the down sides to parliamentary democracies.
The ruling center right party of Prime Minister Pedro Passos Coelho win the most votes, yetit lost it’s majority. So now despite winning, his party has to form a coalition government with other parties. Portugal has seen coalition governments and they tend to be short lived. PM Coelho can form a coalition with left parties, who are against the spending cuts and tax hikes of his party . . .
His party didn’t win the election; they got 38.6% of the vote. 61.4% of the voters prefer other parties.
Given that, it would be an outrageous affront to democracy if that party got to form a one-party government. The requirement to command the confidence of parliament means that they have deal with other parties, find common ground, and construct a platform for government which can command broader assent that the platform they put to the people, and signally failed to get a mandate for.
That’s not a bug, it’s a feature. Building consensus, attracting support from others and making accommodations to do so, etc, are skills which parliamentary democracy fosters. Passos Coelho may have to modify some of the spending cuts and tax hikes he proposed to the people, but as he failed to get a popular mandate to implement what he proposed, is that not a good thing, from a democratic point of view?

Aside from mixed presidential/parliamentary systems like France, Poland and Russia, in what parliamentary systems is the PM not the leader of the party that commands a majority in the House?
If your question is literal :
India has had several prime ministers who came from a party that was rather small. See gujral, gowda, VP Singh, Chandrasekhar
The larger parties could not cobble together support and were willing to support a much smaller one rather than go back to the polls (which might have been inconclusive and expensive)
Such situations were not very stable. Thus the ruling party was not in the majority, but they had the confidence of a majority of the lower house while they ruled.
India is very much in the British tradition as far as Parliament goes, it differs in Federal nature and monarchy…

Well, not exactly. The PM wears two hats, but he is still merely the leader of the largest party. If his party decides to vote against him, it means a new election rather than just a blocked piece of legislation.
Not necessarily. He’s just the person which a majority of parties can back. Traditionally around here he’s also an elected representative, but that’s not a requirement. He’s also not part of the legislative branch, but of the executive. So not comparable to a speaker of the house.
In any case, obviously a king is the best choice.
Ireland has had coalition governments in which the leader of the largest party was unacceptable to the coalition minority parties, so another senior member of the largest party served as Taoiseach instead. (Though this hasn’t happened recently.)
Yes, but I didn’t say the PM was the leader of the party with a majority. The PM is the leader of the party which “commands a majority” in the house. Another way to say that is the PM has the confidence of the majority of the house. The Indian and Irish examples both fit that definition. So did Ramsey Macdonald’s coalition government in the 30s in the UK. He was not the leader of the party with a majority, nor the leader of the largest party. For political reasons, the largest party, the Conservatives, were prepared to be in a coalition with Labour, with Macdonald as PM.
UDS, how did that example work in practice? Did the Taiosech take instructions from his party leader?

Yes, but I didn’t say the PM was the leader of the party with a majority. **The PM is the leader of the party which “commands a majority” in the house. Another way to say that is the PM has the confidence of the majority of the house. **
It’s not another way to say it, these are different things. Your mistake is your assumption (presumably because it’s customary where you live) that PM always means party leader. The PM can have the confidence of the majority party, or coalition of parties without being a party leader, for a variety of reasons.
It could be that the party leader is relatively unimportant, by tradition or due to circumstances. The PM can have been picked as a compromise. A “technocratic” government might be chosen due to circumstances, not led by political heavyweights. The most influential leaders within the party might not be very popular in the population at large, and a less significant but more acceptable name be picked. The main leader of the party might “reserve himself” for a latter election. There might be no clear “leader” of the party due to disagreements and shifting alliance within the party (and the party leader in name not being any of the really influential leaders). The prime minister might have lost the support of the majority of the party while keeping him in office is a better political option than rocking the boat by replacing him. The prime minister can be a figurehead, while other people are pulling the strings. The leader of the main party of a coalition might be unaceptable for other parties in the coalition for whatever reason.
There’s no obvious reason why “prime minister” should be synonymous at all time with “party leader”. These are two different jobs, and whether or not they will be filled by the same person will be dependant of local traditions, political circumstances, etc…
Although in practical purposes both offices are generally combined as parliamentary systems don’t fit that well with divided authority. There was a period in 1940 when Churchill was PM and Chamberlain Leader of the Conservative Party, but Chamberlain’s health was failing fast and at his death Churchill was chosen as Leader.

There was a period in 1940 when Churchill was PM and Chamberlain Leader of the Conservative Party.
That’s a pretty definitive example.
A similar instance, though its Leader of the Opposition and from Queensland where oddity is just another day in Paradise but …
In 2011 Campbell Newman challenged John-Paul Langbroek for the leadership of the LNP in Queensland and won. By political norms this would have made him Leader of the Opposition, but Newman has not been elected to the Legislative Assembly. So Jeff Seeney was elected Opposition Leader while Newman ran the opposition strategy and the LNP election campaign from outside parliament.
About 12 months later Newman won the marginal seat of Ashgrove, the LNP won government in a landslide and Newman took office as Premier.
In the 2015 election in a major swing against a first term government Newman lost his seat of Ashgrove and the LNP lost power in a hung parliament.
Newman resigned from politics, having only ever served as Premier.

UDS, how did that example work in practice? Did the Taiosech take instructions from his party leader?
No. He was the Taoiseach; he had the constitutional status and authority. He was accountable to the Parliament, not to his party leader.
The party leader at the time, Richard Mulcahy, remained in charge of the party organisation, the running of election campaigns and, after the election, the negotiation of a coalition pact with other parties. But because of his role in the civil war, and the fact that some of the coalition parties had been on the other side in that war, he recognised that his seeking the office of Taoiseach would make a coalition either impossible to form, or unstable in existence, so he encouraged his party colleague John A Costello to take that role. Costello was not a front for Mulcahy; as Taoiseach he exercised his own judgment in discharging his functions. Mulcahy served as Minister for Education in Costello’s government.
Costello was designated “parliamentary leader” of the party, while Mulcahy continued to be “leader”. (There was a precedent for that distinction, from a time some years previously when the leader of the party didn’t have a seat in parliament.)
The leader/parliamentary leader distinction was maintained for about 11 years, from 1948 to 1959. During that time Costello served as Taoiseach twice (1948-51, 1954-57) and was recognised for parliamentary purposes as leader of the opposition when not in office.
Both Costello and Mulcahy resigned their leadership offices in 1959, and James Dillon replaced them, reuniting the two offices.
Well, we just had our election, our new Prime Minister takes office shortly.
Meanwhile, the Americans have another, heh-heh, year of campaigning to go.