Corners First
Ernő Rubik's method was Corners First, making it the first ever method.
In David Singmaster's Notes on Rubik's 'Magic Cube', the steps are written as described by Zoltán Perjés.
This is from page 40.
*Not a scan or screenshot. It is a re-typed recreation of the quote.
Zoltán Perjés was a relative of Rubik.
In May 2003, Dan Knights states that in 1999 he saw Rubik solve the cube. He outlined the steps that he saw.
The steps are slightly different here with three edges of the first layer being solved before the last layer corners.
Corners First was used by Minh Thai to win the Rubik's Cube World Championship on June 5, 1982. This is where he achieved the 22.95 seconds world record.
His method was then published in The Winning Solution in 1982.
The first known published Corners First method came in 1978 by Hanke Bremer in Ein Paar Würfeldrehungen.
I contacted Georges Helm about the steps in Ein Paar Würfeldrehungen. He kindly wrote out the steps. Helm's notation is K for edges, E for corners, P for permute, and O for Orient, and the numbers on the left indicate the layers.
Some additional early Corners First methods (1979).
Singmaster, David: Notes on Rubik's 'Magic Cube' - 1979
Bottom layer corners, permute upper corners, orient upper layer corners, solve all edges.
Claxton, Mike: Solution to the Magic Cube - 1979
First layer corners, last layer corners, first layer edges, last layer edges position then orient, solve second layer edges.
Beasley J.D.: A Solution to the Twisted Cubes - 1979
Corners, edges, then centers according to Singmaster in Notes on Rubik's 'Magic Cube'.
Dauphin, Michel: Un Cube Pas Comme Les Autres - 1979
Similar to Bremer’s according to Singmaster in Notes on Rubik's 'Magic Cube'.
The steps for Claxton's method were provided to me by Georges Helm. The others can be found in Notes on Rubik's 'Magic Cube'.