2) Trad (short for "traditional")
Its own adherents and advocates call it "trad", but we shouldn't think of it as the oldest way of roleplaying (it is not). Trad is not what Gary and co. did (that's "classic"), but rather is the reaction to what they were doing.
Trad holds that the primary goal of a game is to tell an emotionally satisfying narrative, and the DM is the primary creative agent in making that happen - building the world, establishing all the details of the story, playing all the antagonists, and doing so mostly in line with their personal tastes and vision. The PCs can contribute, but their contributions are secondary in value and authority to the DM's. If you ever hear people complain about (or exalt!) games that feel like going through a fantasy novel, that's trad. Trad prizes gaming that produces experiences comparable to other media, like movies, novels, television, myths, etc., and its values often encourage adapting techniques from those media.
Trad emerges in the late 1970s, with an early intellectually hub in the Dungeons and Beavers crew at Caltech, but also in Tracy and Laura Hickman's gaming circle in Utah. The defining incident for Tracy was evidently
running into a vampire in a dungeon and thinking that it really needed a story to explain what it was doing down there wandering around. Hickman wrote a series of adventures in 1980 (
the Night Verse series) that tried to bring in more narrative elements, but the company that was supposed to publish them went bust. So he decided to sell them to TSR instead, and they would only buy them if he came to work for them. So in 1982, he went to work at TSR and within a few years, his ideas would spread throughout the company and become its dominant vision of "roleplaying".
Trad gets its first major publication articulating its vision of play outside of TSR in Sandy Petersen's Call of Cthulhu (1981), which tells readers that the goal of play is to create an experience like a horror story, and provides specific advice (the "onion layer" model) for creating that. The values of trad crystallise as a major and distinct culture of play in D&D with the Ravenloft (1983) and Dragonlance (1984) modules written by Hickman. TSR published Ravenloft in response to Call of Cthulhu's critical and commercial success, and then won a fistful of awards and sold tons of copies themselves.
Within a few years, the idea of "roleplaying, not rollplaying" and the importance of a Dungeon Master creating an elaborate, emotionally-satisfying narrative had taken over. I think probably the ability to import terms and ideas from other art forms probably helped a great deal as well, since understanding trad could be done by anyone who'd gone through a few humanities classes in university.
Trad is the hegemonic culture of play from at least the mid-1980s to the early 2000s, and it's still a fairly common style of play. For an example of a fairly well-thought through style of trad by someone who's been influential on the last 15 or so years, check out
S. John Ross's RPG Lexicon.
Both of the next two styles emerge out of problems with trad, especially the experience of playing Vampire (a tradder-than-trad game in its authors' aspirations), but the details of that are larger than this essay can contain so I'm just going to mention it and leave it for another time.