So if you know about more types, and more more datapoints that fit (OR DON'T FIT, that's even more interesting and important!) please share!

Anyway!

Almost all of them seem to have An Inciting Incident, where One Day some supernatural event happened and also a lot of people started "Awakening" into having powers.

The Inciting Incident involves some sort of door/gateway/entrance/void into an alternate space/dimension, from which people-killing-things spew forth if not controlled in a timely manner.

Some follow "first generation" hunters - aka the people on the front-lines are the first batch of people to awaken, but there's definitely works out there following "second generation" hunters (they were kids or teens when the Inciting Incident happened and may be proteges of first-gen ones who are now dead or injured or promoted outside of front-line or retired). SCTIR is Gen 1, but within that the VR dungeon world seems to be Gen 2. Cloudless Sky is also Gen 2

So far I have seen three types

Dungeons - Videogame dungeon instance like stuff which can open anywhere. Hunters enter into the alternate space which has the monsters that need to be fought. Different dungeons are different in terms of environment, monsters, difficulty etc. eg. SCTIR

Towers - Some tower(s) appears and have the monsters inside. Hunters need to go in and keep advancing levels up and up and up, and levels get more difficult in order as you go up. Sometimes you can't come back out once you decide to enter. eg. SSS-Class-Revival Hunter. (I think. I have only read the very beginning of this)

Rifts - I've seen a few different types, but the main commonality seems to be that these generally interact with their surrounding "normal world" in much more immediate way than dungeons and towers, who "spill into" the outside world only if not managed properly. Rifts, meanwhile are much more often a disaster that effect the real world inherently. eg. Hunters Gonna Lay Low, Cloudless Sky, Probably Special Civil Servant too.

Sometimes novels have multiple types of problems, for eg.

Hunter's Gonna Lay Low has Rifts and Dungeons both. Rifts are calamities which appear occasionally, dungeons are day-to-day.

S-Class Chick is Filial (very cute about a 5 year old orphan girl who awakens as a S-rank with crafting skills and has to get adopted by S-class hunters. Very Opposite Yoojin times) has Tower and Dungeons both. The Tower got climbed and "won" a few years in the past, but dungeons still keep popping up and have to be fought back.

The "Awakened People" who fight the threat are called Hunters. Hunters come in two types - Guilds or Civil Servants. Some may have both (like SCTIR and even HGLL).

If it's just Guilds, then Guilds function mostly like video-game / RPG guilds with/without variable levels of cross South Korean company culture. The hunters in the setting are kind of like chaebol/idol crosses.

If it's just Civil Servants, then usually that's mandatory (as in, if you get powers, you are automatically expected to become a civil servant and that's the only way to become a hunter) (Cloudless Sky), or a "secret thing" where the larger public doesn't know about the whole supernatural thing (Special Civil Servant)

If it's both then it seems the Public Servant route is rare and the ones who do take it (especially any S-Class ones) are unusual and overworked and praised. (both SCTIR and HGLL).

I would probably classify Turning by Kuyu, as proto-Civil-Servant type Hunter genre? Like not proto in a meta-way, but proto- as in pure Hunter genre is explicitly a modern world setting, but Turning is IMO adapting the genre for "vague fantasy medieval European setting". And the Cavalry are definitely a government institution, but given the setting there is also a lot of focus on establishing them as a public institution within a monarchy.