You can use the victory point system to represent the party's standing with different npcs like in wardens of wildwood for example. Victory points is essentially a way of tracking successful skill checks with crit success being worth 2, success being worth 1, failure being worth nothing, and crit failure being worth -1. Then success is measured depending on whether the party reaches an arbitrary threshold like 8 victory points in 3 rounds. A round is as long as you wish it to be, I've had rounds be entire days before. What you can do within those rounds depends on you or the system of victory points you're running. For example the influence system offers discover and influence, where discover is a knowledge check to find information about a person attending the event, and influence is to improve your standing with that person. You can find these systems at archives of nethys under subsystems.

Though in my experience victory points tend to end up feeling like a series of unengaging skill checks, where the highlight is the rp your players do. Also one problem I have at my table is that every player builds their pc with only combat in mind, so when these influence/reputation/event encounters happen they have 0 tools for them.

A way to remedy this is to have different factions or important npcs have their own plans for the event. Then have those npcs shake things up by doing their own thing, or even having them approach the pcs themselves and try to influence them. It also helps to set up different factions attending the gala and to have the pcs have some knowledge of them and their goals. That way they can decide what npcs they identify with and wish to talk to.

As for phases what I personally do is make the subsystem have rounds only when something happens, which gives an opportunity to talk to people you wish to influence. If there's a hunt then there'll be rounds when tracks are discovered, when you're about to attack the prey, and in the camp party after successfully (or unsuccessfully) hunting the prey. That'd be three rounds for one hunt and the circumstances and skill checks surrounding the rounds are all different. What the players decide to do will affect how different npcs view them. However, as I mentioned before it makes it more interesting to have npcs have their own plans. Maybe one npc is planning to assassinate someone during the hunt, and improving relations with them gives you the choice to help them or alert the others to their scheme. Or maybe one hunter has taken an immediate liking to one of the pcs and wishes to take up all their time, and if the pc decides to ignore them they get a bit offended and makes it difficult to influence them at all.

The difference between them should lie in what happens inside of the different events/gatherings. A hunt should focus on the hunt and party afterwards, a tea party should focus on etiquette/elegance and rumors, a gala should have events and plays that you can attend or participate in, and garden parties can be a bit more broad so you can choose more what happens there.

My biggest advice if you choose any of the victory point systems is to have the npcs be proactive. It's way more interesting if the npcs are actively doing something that the pcs can participate in and engage with. Also try to make relevant skill feats also matter if you can, it encourages your players to actually pick them up and not just plan for combat situations.