Authorized instructor is defined in part 61.1.
Authorized instructor means—
(i) A person who holds a ground instructor certificate issued under part 61 of this chapter and is in compliance with §61.217, when conducting ground training in accordance with the privileges and limitations of his or her ground instructor certificate;
(ii) A person who holds a flight instructor certificate issued under part 61 of this chapter and is in compliance with §61.197, when conducting ground training or flight training in accordance with the privileges and limitations of his or her flight instructor certificate; or
(iii) A person authorized by the Administrator to provide ground training or flight training under part 61, 121, 135, or 142 of this chapter when conducting ground training or flight training in accordance with that authority.
In the airline world, many times the "instructor" does not hold a CFI certificate. But the airline authorizes them to instruct their pilots. In your example, the "authorized instructor" for part 61 is a CFI.
As to your first question, I hold a FAA CFI with SE, ME and instrument airplane ratings and a glider rating. My first aviation job was with a fixed base operator (FBO). It was a company that operated an airport for the city. They sold fuel, rented hangars, did airplane maintenance and provided instruction under part 61. People could call and say they wanted to learn to fly and the FBO would rent them an airplane and schedule an instructor for them. The person would pay the FBO and I would be paid by the FBO. But the FBO did not have an official syllabus approved by the FAA. We only had to make sure all the requirements listed in part 61 were met. Currently I spend some time at a glider club. There are several members who hold glider instructor certificates. The club has a syllabus of what items should to be covered in each lesson. But it is the club's own idea of how a glider pilot should be taught. There is no FAA approval. Again, instruction under Part 61.
If you have your own airplane or glider, I could provide instruction independently from the glider club or any airport flight school. As you said, conducting the ground school and briefings "under a tree."