Two organizations run Formula 1, and most people only know the name of one of them.
The FIA governs the sport. Formula One Management, which is owned by Liberty Media, runs the business. Rules, stewards, safety and licenses on one side, and TV deals, race hosting fees, promotion and branding on the other.
The FIA does legally own the Formula 1 World Championship. The commercial rights were leased to Formula One Management through 2110. So Formula 1 is owned by its governing body and operated by a media company.
Now what the FIA controls is technical regulations, sporting regulations, and financial regulations. Circuit safety grading, super licensing, and the stewards. That covers most of what's come up this season: parc fermé, the two-compound rule, ADUO engine upgrades and penalty points.
As for how the stewards work, there is a different panel every weekend, including a rotating driver steward. They work under time pressure with limited information, and teams have repeatedly asked for more consistency, and the rotating panel is a part of the reason why.
The Concorde Agreement is the private contract binding the FIA, FOM, and the teams. It sets how revenue is divided and how the structure can be changed. No one writes the rules on their own.
The FIA is the referee and F1 the company is the promoter.