Thought I'd cross-post this here, just to gather some attention to my timeline over here:
https://www.alternatehistory.com/forum/threads/the-world-born-at-tilsit.413111/
Constitutional Monarchy:
Mostly the same as OTL. The Monarch has no true power, rather most of it is invested in the Parliament/Congress/various other publicly elected council. There's generally Universal or near-Universal suffrage among their citizens.
Semi-Constitutional Monarchy:
Similar to Constitutional Monarchy, except that the Monarch has at least some true power. Degrees of power range from nearly-absolute to being only able to declare a national holiday.
Aristocratic Monarchy:
Similar to Semi-Constitutional Monarchy, except that suffrage is limited to only the land-holders, or other such requirements. The Parliament is usually made up of lesser nobles. Like Semi-Constitutional, the power of the Monarch varies vastly, from near-absolute, to trinket (such as Constitutional)
Absolutist Monarchy:
The Monarch's rule is law, no other force can legally change or go beyond the law set by the Monarch.
We also have governments that only exist in one country (or, at least one country on the map).
Imperial Republic:
While the names and titles of the "Imperial" Republic often makes people think that this is a type of Monarchy, it actually has more in common with a Republic than a Monarchy. The position of Emperor is for life, with the position of Prince elected upon every 5 years, starting with the year the previous Emperor died (or early the next year, if necessary).
Russian Republic:
Based heavily off of the American Republic, the Russian Republic has two chambers of Legislature, the Congress of the States and the Congress of the People (although these are of equal size, featuring 1 representative from each state). The Representatives of the People are elected directly by the people, while the Representatives of the States are selected by the Governor of such state. The President has similar power to the American President (as of 1914), although slighter more powerful. There's also a Judicial court to make sure everything is legal, constitutional, and in the favor of the people.
(Rhenish) Elective Monarchy:
In many ways a continuation of the Holy Roman Empire, the Prince-Primate is role chosen for life among the the Kings, Dukes, and Princes within the Confederation of the Rhine. This Prince-Primate controls most of the day-to-day federal business of the Rhine (however little there is), although if the Protector (who is also Emperor of France) dislikes a choice the Rhine does (including their choice of Prince-Primate, sometimes even retroactively), they can choose to veto the Rhine's decision.
Note that I wanted to bring up here: Essentially since the death of Napoleon I, there's been a 3-way power struggle over the Rhine between Westphalia, Saxony, and Bavaria. Westphalia inevitably gets most of the power because they're more closely connected to France, but when the spats between Bonapartes (or Bonaparte and Imperial Republican) happen, Saxony often gets most of the attention as France's favorite pet in the Rhine. Bavaria often tries to show off their splendor and power, but are ignored by France, but do get a lot of attention from other German Monarchs.
Swiss Parliamentary Confederation:
Central power within the Confederation is essentially limited ability to declare war. Armies and everything else is dependent on the cantons. It's usually up to the French Emperor to make sure that the cantons play nice with each other.
Greek Triarchy:
After Constantine Ypsilantis successfully liberated Greece (with Russian and French aid), the Greek government was set up as the "Three Pillars of Greek Philosophy:" Byzantine Monarchy (hopefully with more stability), Byzantine Theocracy (minus the meddling of the Monarchy), and Athenian Democracy. As such, Greece has three executives, of which 2 have to agree upon an act for it to follow through (originally, all 3 had to agree, if one voted no, the resolution was out. This turned out to be a bad idea).
Greece has 3 houses of parliament. The House of Nobles, the House of Priests, and the House of Citizens. Because of the fact that there were nearly no nobles at the beginning of Independent Greece, back then, and indeed today, the majority of the house of Nobles was made up of Generals and Merchants, those that bought their way into the House. The House of Priests is made up of the Clergy and generally selects itself, and the House of Citizens is directly elected.
Turkish Republic:
Nothing really to say. Just your average unitary republic.
Patriarchal Theocratic Diarchy:
The Patriarchs of Antioch and Jerusalem share power within the Two Patriarchies (waiting for the other 3 Patriarchies to join to complete the Pentarchy once more). The Patriarch of Antioch has absolute dominion over his section, while the Patriarch of Jerusalem has absolute power over his. It requires agreement between both for any diplomatic (or foreign militaristic) actions to be done, but the Two Patriarches are usually in agreement with each other (it does help that when one dies, the other can essentially choose the other's successor, or at least heavily influence the nomination).
A couple of other governments I want to talk about
Albania:
Albania has, like Greece, 3 houses of parliament, the Councils, but instead of one for Nobles, Clergy, and People, they're instead for Catholics, Orthodox, and Muslims. The Prime Minister is technically supposed to be the head of all three councils, but usually has to gather influence through the de facto leaders of the other two councils. The Prince thus needs to take a good amount of power for themselves.
Austro-Bohemia:
Boy what ton of shit Austro-Bohemia is in. Although have one executive, the country essentially runs as a Diarchy that despises the other half. The Bohemian half is a Constitutional Monarchy that wants as little as possible to deal with the Monarch, while the Austrian half is Aristocratic Monarchy that has the Monarch have a lot of power, with the Chancellor essentially being a puppet. Thus, for anything to me done on a federal level, the Prime Minister of Bohemia and the King-Archduke have to agree, which often can lead to deadlock.
Thuringian and Anhalt Unions:
Basically two mini-confederation already within the Confederation of the Rhine. Most of the purpose of these unions is to decide how to vote on the confederate level, such as choosing a new Prince-Primate. So now there's a voting block of 3 and 8-11 Princes on the College of Princes that one needs to influence to get any sort of legislature (what few there is) through. I should mention as well, a couple of other, smaller princelets were annexed or united into larger Duchies and such.
edit: Somehow forgot to occupy Morocco.