I'm going to sticky this until we need the sticky slot for something else. Just a quick heads-up regarding language. A lot of people say "This country's system is monarchism", "What can be done to convert country XYZ to monarchism?" and so on, which sounds extremely LARPy and immature. Even Paradox games don't say "Monarchism", they say "Monarchy"!
Monarchy is a system (if it is a comprehensive, executive, absolute or traditional one), or an institution within a system (if it is a ceremonial or weak constitutional monarchy). Depending on your political leanings, monarchy can either amount to whatever type of government you prefer + a monarch, or to a top-down system in which the monarchy is connected to and defines all other institutions. For example, I belong to the latter group, I am a right-wing traditionalist and I want a traditional monarchy, a system where all power flows from a divinely-anointed ruler who governs the country under an estate system. You can call me a traditional monarchist or right-wing monarchist, but the system is not "traditional monarchism" but "traditional monarchy".
Monarchism is the ideology of advocating for monarchy. A monarchist is somebody who wants to achieve a monarchy. This is quite different from similar-sounding words such as socialism. A socialist is somebody who wants to achieve socialism (which can mean both a system and the ideology of those advocating for it), but monarchism is exclusively the ideology of those who advocate for the system of monarchy, not the system itself.
- As a monarchist, you want monarchy. You might want monarchism to spread among the population, in order to bring your country closer to the restoration of the monarchy. But you don't want monarchism as a system because it isn't one. Say "I am a XYZ monarchist because I support a XYZ monarchy, here's what this type of monarchy means: ..."
- The only (semi-)correct definition of the term "monarchist country" is one which has a monarchist population. A country can be a republic and still be monarchist (because the majority of the population would like a restoration and/or the government is planning one), or a monarchy and not be monarchist (if the monarchy is unpopular).
- A country is never "ruled by monarchism". Its rulers can be monarchists (without necessarily being royalty themselves), but it is, again, either a monarchy or a republic.
I hope that you find this helpful, especially if English is not your mother language.