Sure, it's what he started as (and viewed himself as), but the idea of a separate Protestant identity emerged very quickly. Whereas it really didn't work that way in the CK period (at least, in the West; the Orthodox church did have identifiable Iconoclast and Iconodule identities). And the printing press was a gamechanger in general, which doesn't exist in the CK period, and was critical to building this sense of broader identity (as Protestant and anti-Protestant propaganda could be distributed cheaply and widely in a way that really wasn't possible earlier).
I feel like the the early protestants viewing themselves as separate from the Catholic comes from the Catholics calling them Lutherans (a naming convention they got from previous heresies) and the local nobles using the religious movement for their own political agendas. The printing press just helped with speeding the whole process up.
The actual teachings of Martin Luther and his Lutheran followers, while taking criticism of Papal authority, also refer to themselves as the Catholic (or universal) Church. They viewed themselves as going back to the roots of Christianity and that the modern church had lost it way. These are all qualities that many medieval heresies had. So I fail to see how the Lutherans are different than medieval Catholic heresies, besides success. It really seems your criteria for if they had separate identity is success. In which case every heresy is capable of that (though some might have a harder uphill battle).
Finally, but I think most important, what does it matter if they had a separate identity or not? The main church viewed them as such, by branding them heresies, so it makes sense from a game point of view to follow suit. And while it might be interesting if CK3 could somehow capture the nuance of how heresies go from just a difference of opinion to being expelled from the church, it is probably easiest from a game point of view to have them as separate faiths from the get go.
Lots of local mystic or reform movements that were accepted by the church had heterodox theological ideas that were accepted just fine. Catherine of Siena, Francis of Assissi, Clare of Assissi, etc. Even the Waldensians weren't initially condemned as heretical. The church was generally just fine with these sorts of movements, as long as they didn't threaten the hierarchy (so the more extreme Franciscans were later classed as Fraticelli and condemned, but the Franciscans themselves were accepted as not just perfectly orthodox but a pillar of the church).
I feel this is moving the goal post a bit. You characterized medieval heresies as being about going from "'let's give more of our money to the poor' to 'hey, maybe the local bishop should give some more of his money to the poor as well'". But when me an others point out there are medieval heresies with beliefs beyond church wealth (and Papal authority), your response is only the ones that "threaten the hierarchy" were declared heresies. Which yeah, the Catholic Church had a political agenda. But this gets us back to where we started, because the reformation movements fall into that same description. And so again what is separating them from medieval heresies besides success?
And transubstantiation wasn't nearly the touchstone in the medieval period that it would become later. It wasn't defined as dogma until after Waldo had already died.
Waldensian also survived Waldo's death. So I fail to see the point in this.
Also while it might not have been official dogma until 1215, that didn't stop church citing the Petrobrusians lack of belief in transubstantiation as one for the reasons for declaring them heretics in 1139.
Sure, but most of the "nature of Christ" heresies were mainly pre-CK3. Arianism, Monophysitism, etc. had all broken out well before then.
True, not the best example of difference in beliefs for the CK3 period. Just having studied early Christianity, it always the one that comes to mind first.
The CK2 period was absolutely a time of spiritual flowering and religious movements in Catholicism (I admit to not being nearly as well informed about other religions during this time period), but to define it as a time of rampant heresy and constant crises shaking the church is frankly ridiculous.
When have I defined it as a period of rampant heresies? I just have said that larger faith seem to have a larger problem with heresies than smaller faiths due to their size, thus I like that the fervor mechanic represents this. But I have also said I think the fervor mechanic could be improved. I have also said that the manner in which heresies have spread CK3 is wrong, but that while there general severity should be reduced, they should still have the potential to become something serious (again mentioning needing the top liege to fail to combat it while at the same time the local nobles choose to embrace as a vehicle to rectify their grievances).
I've also said that this should only play a part in showing the conflict within Christendom, with councils and papal authority also being important.
So no where have I argued for rampant heresies. Just that Catholics should get them, and that there be a possibility of them being serious (essential an early reformation). Doesn't need to be very likely, but it should be there to encourage Catholic rulers to try and nip heretical movement in the bud.
The various reform movements generally saw themselves as part of the church, they rarely wielded any sort of political power (local nobles might have some sympathy with them, or be accused as such, but generally were willing to abandon them once the main hierarchy condemned them as heretical).
I feel this is again the problem of using the measure of success to define what really counts as different or not. And again it doesn't really matter because the Catholic Church declared them as different.
Also sure plenty of nobles stopped their support for them when declared heretic, but there are also example of nobles continuing afterwards (Lollardy comes to mind as one such and Abbot Bartholomew protection of Angelo da Clareno after he was excommunicated). This again mirrors the reformation, expect on the level of how successful they were (and thus how many eventual did it).
It's actually enormously debated among historians as to what extent Catharism even existed as a movement, versus being essentially a paranoia among local inquisitors. Everything we know about it comes from the records of inquisitors, who generally asked formulaic questions and decided whether they believed the answers. So they had a checklist that they'd assembled (based mostly on hearsay and previous interrogations) of "signs of Catharism," interrogated folks to determine whether they met any of those signs, and if they decided that they did, labeled them as "Cathars" in the files.
I agree that its existence is debated (along with plenty of other heretical movements). But regardless the persecution is something that happen (whether the foe was imagined or not). So the fact people might have been wrongly accused shouldn't be a reason for not having the "cathar crusade" in the game. I could see an argument for the manner that it appears in game to be changed. But there should be a system for allowing Catholicism to have these internal struggles, whether against real or imagined heresies.
Similar to how witches aren't a real thing, but we need a way for them to "exist" in game, as that was a real concern of medieval people.
Also, since it is debated, Paradox might as well do the more entertaining thing and make Catharism a faith players can play around with (just need to tone down the ai).
So again, not saying the CK3 system is perfect, but heresies (and especially the fear of them) I think are important enough to be represented in CK3. How they are represented could be improved. And for the purpose of this thread, I think they should be more likely in large faiths, which the current fervor system does (though the current values should at least be tweaked, if not the system being revamped).