The most common word would be a university, college or academy. Note that the word arcane which you use literally means "hidden or secret knowledge", and was used for mundane scholars in early Europe, to describe things which they studied and practiced. The concept of naturalism is modern - it has an ancient history but came to prominence as "opposite of magic" during the enlightenment. So in a medieval setting there would not be much distinction between learning magical knowledge and learning mundane knowledge - without the scientific method and a reasonably mature theory of physics there's not a clear distinction between the two. People would care more about the fact that it's a place where people are learning some uncommon (i.e. arcane) knowledge, so the places of study of magic would be called universities, colleges, schools and academies.
Following real historical tradition, I would expect a university to include several distinct areas of magic, or perhaps combine magical studies like conjuration and alchemy with mundane such as math, rhetoric and history. I would expect a college to house a group of peer or near-peer magicians who interact (whether through collaboration or competition) in their studies. A school or academy should have a clear distinction between a senior "teacher" class and a junior "student" class, with some students being destined for the former while others expect that upon graduation they would rejoin mainstream society with their skills and knowledge.
A laboratory, sanctum or tower invokes the sense of an individual wizard, or small group, working on some project with little concern for educating the future generation of magicians. It could also serve as a workshop or place of business for a wizard who provides services to the general population, similar to an architect's office or a blacksmith's shop.
Towers are preferred by political leaders, such as heads of magocracies, because it is a conveniently exotic counterpart to a mundane lord's keep or fortress, and is a bit like the Vatican palace. Witches like huts in wooded wilderness, often haunted.
A conclave, cloister, lodge or guildhouse is what you should use if you want to emphasize that the wizards try to isolate themselves from society, perhaps they even operate in secrecy like a cult. This is different from universities and academies trying to create an environment free of layperson riff-raff so that scholars could learn in peace. A university's doors are always open and it's no secret what they do. Much like a library, the point is not to keep people out but to maintain a specific environment. So long as people conform to that environment they are welcome. With a conclave or cloister, I would expect that outsiders would be explicitly kept out and/or kept in the dark and seen as undeserving of the secret knowledge. For small groups, "hermitage" might also work. However, in your case it sounds like the location acts as a place of learning with general admission, so these terms would seem inappropriate.
I wouldn't use something like "monastery". The point of monastery is for the monks within to quietly contemplate their faith, introspect, pray and minimize interactions with both the outside world and their fellows (many monasteries discourage chitchat and excessive socializing between brothers). This makes sense in a religious context, but for magical study it would be counterproductive, unless your setting's magic is very intuitive and based on ineffable inner strengths and wisdom which transcends words.
Everything else is a fancy neologism, because it's used much less often in popular fiction, so people won't respond to it as a widespread technical term. If you don't want to use any of the above, you might as well just invent whatever word you think sounds cool, without worrying if it has much precedent for use.