The modern view of a magician is someone who gains innate access to supernatural powers after a long period of practice and study. Call this 'Magic by Training.' This is the magic featured in most movies, most books written after The Lord of the Rings, and in most computer games and role-playing games. I use the term 'magician' in a very broad sense here, so the term would apply to Luke Skywalker as well as Harry Potter.
The mechanics of the training are different. Sometimes it is more physical, and sometimes it is purely mental. Sometimes magic is tied closely to emotions, and sometimes it is as dry and mathematical as an algebra problem. Sometimes anyone who studies hard enough can do magic, and sometimes it takes a combination of innate power and study. What all the representations share is that, after enough study, the magician has direct access to power, and can wield it just as readily as a cowboy can pull out his six-shooter.
There is some historical precedent for this. Many Oriental cultures believe that people have a life-energy, or ki, that can be harnessed for magical powers through training. There are many Jewish legends of rabbis gaining magical powers after long and dedicated study of the Torah and Kaballah.
However, the idea of magic in European folklore, and in most cultures throughout the world, is nothing like this. A European witch or sorcerer or magician was someone who summoned demons and made deals with them. The only study involved was looking up the name of the demon you wanted to summon, the proper ritual for calling that demon, and ways of bargaining with the demon. The magician might be granted one or two powers by the demon, but those powers were always tied to the demon. They never actually belonged to the magician. For example, all of the folklore on witches said that they must have a 'familiar', or demon in the shape of an animal, that stayed close to them and gave them their powers.
Magic in other cultures was very similar, except that the magician would call upon ancestors or nature spirits or one of a pantheon of gods. Most 'magic' throughout human history is best described as 'prayer' rather than spellcasting. Roman magicians made appeals to gods and spirits. Tribal shamans asked Rain to come and visit for a while, or asked their ancestors to bless their houses. Even in the case of rabbis or Buddhist monks, the legends often imply that God or Buddha gave them their powers as a reward for a good and virtuous life.
Call this 'Magic by Politics'. For most of human history, magic meant making an appeal to a sentient supernatural being, and asking that being to use its powers to aid you. The idea of gaining powers yourself was very rare. Magic was who you knew, not what you could do. The magician almost never had any kind of immediate practical power. Everything was prayer and ritual. The magician would spend a few hours making contact with some kind of thing and bargaining with it so that it would do what the magician wanted.
When the Harry Potter books came out and were attacked by fundamentalists, I remember thinking that a witch as described in the Bible was nothing like a witch at Hogwarts. Whenever the Bible talks about witchcraft or sorcery, it is always describing people who are making with pacts with demons. Harry and his friends never do anything like this. Their magic is an innate and technical skill, like taking the derivative of a mathematical function.
Despite the trappings of tradition, the metaphysics of Hogwarts magic is thoroughly modern. No traditional European magician would recognize any similarity with it. This portrayal of magic could only exist in a modern, educated, industrialized culture where people are expected to go to school for years to learn the technical skills necessary to function in a complex and artificial world. There is very little mysticism or magical thinking or spirituality at Hogwarts. It is Magic by Training, and mastery of the occult is made to look like an algebra class.
The closest modern equivalent to traditional European and Christian views of magic is a drug deal. You are seeking contact with dark and mysterious powers in order to obtain a forbidden thing that brings immediate reward but will almost certainly ruin your life. The medieval warlock sought the names of demons and their summoning rituals the same way that modern druggies seek the names of good dealers and the location of the street corner they hang out at. In other cultures, magic and the things you deal with to do magic are less obviously evil, but they are often mysterious and alien. It was definitely Magic by Politics, and the process resembled a trade negotiation much more than an algebra class.
In Jonathan Strange & Mr. Norrell, this dichotomy is made clear. The titular English magicians are doing Magic by Training, and they are good at it. Mr. Strange can, for example, build a temporary road from one city to another with about as much effort as a potter would use to make a vase. He does not call upon spirits of earth or stone, he just does it with willpower and technique. By contrast, their fairy antagonist does nothing but Magic by Politics. He has made deals with entities like the Dawn and the North Wind, and all of his magic consists of asking them to do things for him.
I believe that society's view of magic has changed because our economies have changed. Before the Enlightenment and Industrial Revolution, the only reliable way to gain wealth and power was by politics. If you wanted to get anything done, you had to know the right people and make deals with them. Gaining favor at a noble's court was a much better way to get wealthy than learning some kind of skill. Wealth was not something you made, it was something you got from other people. Their views of magic reflected their views of life: you got things by making friends with the right people.
But after the rise of the modern economy, the best way to gain wealth was by training in a technical skill. We still have corruption and patronage, of course, but it is much less pervasive than it was before, and you often need a technical skill before you can benefit from it. Our modern view of magic reflects our modern view of life: the best way to get wealth and power is to spend years studying a difficult skill. Once you master this skill, you have knowledge and abilities unavailable to most, and these innate abilities let you do what you want. Dumbledore is much more like Steve Jobs than Albertus Magnus.