Overrated: La La Land is a technically accomplished film about technically accomplished people being mediocre
On the question of what a great film performance looks like from the inside, based on how directors have described watching their actors work.
The most useful description of what distinguishes a great film performance from a good one comes from directors who have watched many takes of the same scene. The good take is technically correct, emotionally legible, and covers the script's intentions. The great take does all of this and also reveals something that no one, including the actor, planned.
Aleksander Sokurov, talking about his work with his actors in films like Father and Son, described watching a performer's face change between the line readings and seeing something authentic break through that the performance had been suppressing. He printed the take between the lines.
Kubrick reportedly shot some scenes of The Shining for months, accumulating hundreds of takes, looking for the moment when the technical control of the performance broke down enough that something genuine emerged. Whether this was the right method for Shelley Duvall is a separate ethical question.
What both accounts suggest is that great film performances involve a collaboration between preparation and accident: the preparation creates the conditions, and the camera's patience with time allows the accident to occur.