I got into ArM with third edition back in the 90's. It was also the time when the original Vampire: The Masquerade was released and both games had a major impact on my roleplaying life, each for different reasons.
I think that the idea behind V:tM, that there is a "secret history" of the world; that things appear normal on the surface but the real underlying causes for major historical events were actually guided by a cabal of ancient, hidden creatures really appealed to me.
This idea influenced my implicit understanding and interpretation of Ars Magica: Mythic Europe is a place that could have been our historical past but the "truth" (that wizards, dragons and fairies, etc. really existed) has been forgotten over the past 800-odd years, maybe even by design. Anyways, what it meant was that my Ars Magica has always been pretty low-powered with magi leaving only feint footprints in history. Over the years I have always assumed that the Order was a hidden, secret organization known only to a few important church members and selected nobles. Magi had to make subtle arrangements to build a covenant on the lands on a lord and had to find a suitable cover for their activities. In case they were discovered, the understanding was that the locals would "not suffer a witch to live" and take action appropriate action against the coven of sorcerers in their tower on the hill.
This implicit assumption and sub-conscious interpretation of the various ArM supplements was never really challenged until a relatively recent batch of books: Hermetic Projects and Transforming Mythic Europe. It made me realize that, perhaps, the default setting (and I know that there is no one-true-way to play ArM, but there is still a set of default assumptions baked into the books) was way more fantastical and high-fantasy than my original understanding of the setting. I have been re-reading some ArM5 books while preparing for a new saga, and I can now totally see how I missed some pretty obvious things describing a far more fantastical setting than I've always understood it to be.
So, I'm curious, just how fantastical is your Ars Magica Saga? Do you have castles that appear out of nowhere overnight and become home for a new group of wizards? Does the local noble realize that some of those "those strange Order of Hermes folk" will start making a nuisance of themselves and he better make a plan to accommodate the new arrivals? Does the local priest tell his flock that the wizards with their clockwork servants and floating litters are nothing to be feared? Do the local peasant's petition the magi to save their harvest by calling in some long overdue rainstorms?
The things you describe are not exactly mutually exclusive. I, personally, play in a setting that embraces the Jerbiton chapter from Houses of Hermes: Societates, so while the power of magi is almost never felt by anyone outside the Order and their own grogs/peasants, the higher parts of society (nobles and clergy) largely recognize the Order's existence. Even so, the closest to interaction the Order actually gets with most people not directly affiliated with the Order are strange occurrences happening a ways away which the Church either takes credit for or blames demons for depending on whether it's a good or bad thing. And occasionally a weirdly dressed and usually untrustworthy man taking part in a feast, with few or no supernatural occurrences accompanying it.
Again, though, the two options aren't mutually exclusive, largely because there are effectively two worlds at work just when evaluating the magical highness of Mythic Europe in relation to the Order of Hermes. I mean, plenty of magi, especially when working together (though many can individually), are in fact capable of erecting giant castles with magic and weaving great defensive enchantments over them, and other similar feats that would wow mundanes, but since they do so within the Magical Misty Forest of Cliche Names, the mundane world doesn't notice and goes about its business in just as seemingly low-magic a world as they started in.
... Now, I don't find Ars Magica 5th Edition altogether low-magic even if the Order of Hermes keeps entirely to itself. Because it's a world where everything people believed in back then was true. Even without the Order of Hermes anywhere to be seen (or unseen, for that matter) there are still witches and magicians forming groups and casting spells for the towns' or their own benefit, parish priests calling miracles that heal the injured and cure the sick, and Faerie nobles leading armies into towns and forcing everybody to retreat into the church. There are a LOT of subtle things in Ars Magica, don't get me wrong. Shapeshifted demons tempting people, a hidden Order of powerful wizards, and mystical places filled with powerful creatures that nobody actually comes into contact with because "the misty forest is creepy and the church told us not to go." But there are also a whole lot of really obviously supernatural things going down, some of which even peasants interact with on a regular basis. So I'd call Ars Magica solidly mid-fantasy; it's not exactly high-fantasy, because while supernatural power is pretty much everywhere and is well-known (excluding it from low fantasy), most of it seems weak (at least on the surface) and people wouldn't fare a whole lot better, worse, or even differently if it didn't exist (making it hard to call high-fantasy).
In general, I assume that the Order's existence is (a) common knowledge and (b) a headache that most nobles and clergymen don't really want to deal with. Simply put, smart noblemen know better than to pick fights with covenants, because the Order can easily knock down a few petty nobles who want to pick fights with them, and kings usually have better things to do.
Magi don't need to be secretive, because this isn't V:tM and mortals don't have the advantage. The Code of Hermes is there because the magi who make the rules don''t want to be involved in petty little power games with mortal kingdoms, they want to sit in their labs and expand the outer limits of magical power. Lords of Men has a list of reasons -why- history has more-or-less run on schedule up until now, and I prefer "The Code Works As Intended." Until your PCs start wrecking things, Europe will run mostly on historical lines.
My take on why things are as they are is that the magi have all of the power and none of the responsibility. The code of hermes tends to support this state of affairs by allowing the magi say "sorry I can't get involved in your stuff, I have 1000 magi sworn to kill me if I do"
Certainly the magi can help their neighbors if they choose failure to do that is just being a crappy neighbor (and may itself "bring ruin upon their sodalies") but if they let themselves be forced into helping then there are possible complications. Complications are good for stories, I'm not sure that the code we have is the best tool for forming stories, but my gut feeling is that it is more conducive to it than a code of secrecy (I'd be willing to look at changing the setting to promote even more interactions).
So I'm pretty high fantasy in my games.
I think fifth edition did a good job of taking what was often thought of as a continuum between high fantasy and high history and breaking it to give us both more fantasy and more history.
I find the setting an interesting juxtaposition. On the one hand, it is extremely fantastical for a "real world" setting. Magical castles, archangels, 1200 magi in a big orginzation in europe.
On the other hand, it is also extremely consistent with what people actually believed at the time to be true - platonic ideals and all.
Personally I prefer things rather less Fantastical, as too much fantasy detracts from the enjoyment of exploring the world of medieval Europe. If I were to redesign the setting I'd ditch the current concept of the Order of Hermes and have a much smaller universe of magi interacting much more closely with the mundanes of the world.
The stuff that people in the middle ages believed in was pretty fantastical.
Castles built overnight by faeries, witches, or the devil himself? Check!
Flying dragons that were regularly sighted? Check!
Dead folks raised to life or unlife by holy men? Check!
People flying to the moon or digging their way down to hell? Check!
Now, it's true that the most conspicuous stuff happened "a few generations back" or "a few days/weeks/months travel away", so the more "local" the story, the subtler its supernatural elements tended to be (the strange girl in the cottage at the bottom of the valley would probably "only" talk to animals, or curse men with impotence); but there's no contradiction between "a world just like the medieval folks thought it was, and that we have forgotten/rewritten" and the way it's depicted in ArM5. At least in my opinion.
I do not feel high fantasy at all detracts from the setting. If you want to "explore" Europe in the middle ages, get a history book. I myself go for, what some would say, a "high power" setting. I disagree, for I have yet to see any player character do anything world shaking or history altering at all. As for meshing with history, I am of the opinion that magi are part-and-parcel of human history. Some stuff in history may have been caused by magi, a-la VtM style. No conspiracy, just alternative history. And some stuff in Hermetic history is affected by the mundanes. It is not a separate history. Magi and mundanes fall into the same category as far as I am concerned. They are human beings part of human history, and they all affect and are affected by human civilization.
I think there's a difference between what people actively believed and stories they told each other about fantastic things. If people actually believed all that fantasy, to a level of belief which determined their everyday actions, they wouldn't have acted anywhere near as rationally as history suggests they did (which is not to say 100% rationally, but not notably more irrationally than moderns).
We are talking about a time where everything in the bible and other holy texts was taken as 100% written truth. Flying dragons and castles built with magic are pretty tame in comparison. As for the real 1220 and people behaving rationally... well I think nobles and churchmen were a good deal more rational than the common folk and their actions and decisions largely shaped the overall cource of history.
Pagans and atheists existed (as a minority that by necessity kept itself hidden, but still) and I was under the impression that religious people today believe in their holy books in full except for the listed times of happening.
That said, yeah. Common people as a majority tend to believe almost anything if it's force fed to them hard enough. Of course, as mentioned in the core rulebook and can be confirmed with historical reading, magic was generally condemned as lies and trickery by the Church in 1220, so this might not help Ars Magica's case.
I can't speak for Islam or Judiasm, but in Christian circles, taking the Bible as literal 100% truth is a function of relatively modern Protestant American fundamentalism - at best, that particular strand of theology goes back to the late 1800's.
The best description of my saga would be "Discreet high fantasy": there are abundance of faes, magical creatures and such (including Baba Yaga and Zernebog), mages have access to pretty powerful spells and are about to build a flying "base camp" (a house with two labs to be able to study interesting sites with a lab nearby).
So far, they took every steps to remain hidden and discreet, but there are more and more supernatural incidents happening and since it is a modern setting (Russia present days), it is only a matter of time of those "outrageous and naive stories" become too numerous to be discounted and only reported in fantasy newspapers.
The Orthodox Church and a few governement offices are aware that "some serious shit is about the hit the fan".
In parallel, one mage is heading a political party "Nature and Tradition" heading the neo-slavic / pagan renewal, with the goal to be ready to negotiate early on integration of surnatural creatures (and magus) in the society.
Maybe in the far north, beyond where Christianity could be bothered slogging through the snow. But then again, it's an area where very few people at the time bothered to go, so that's entirely understandable.
Scandinavia was (mostly) christian by 1220 AD, certainly the scandinavian countries were all officially scandinavian by 1220 AD and had been for years (well over a century in every case IIRC).
Admittedly, this does not automatically mean that everyone was christian, but the overwhelming majority were, and practicing pagans were probably being very quiet about it. Indeed the danish king embarked on "crusades" against Rügen and the baltic states (as mentioned in GotF and the Dragon and the Bear) at this time.
You'd have to go fairly far north to find open pagans I'd say.