When 5.0 first came out in 2014, the monster manual included a satyr for use as an NPC. Despite wasting tons of space, you can see how powerful this description is. 5.0 may not be the best at this sort of thing, but man is it up there. You read this and you know what a satyr is supposed to be, and it's compatible with the faun-evolved version of the mythological satyr that you would expect.

https://drive.proton.me/urls/MBSHZBWPVW#qMduUcy79dmS

If you look into this statblock, you see the main mechanical elements of the 5e satyr. First, he has the monster ability "magic resistance", which grants advantage on saving throws versus spells and magical effects. When the PCs are interacting with a monster, it's usually very clear which of their effects are magical and which are not (if your magic sword made the effect it is magical, for instance). We can also see that he has a move of 40, standard for things that are faster than a man over tactical distances, such as a lot of creatures with animal legs, be they prey or predator. His performance skill is about as high as a low CR statblock can be (he's built with expertise in performance, adding his proficiency of +2 twice). He's noted as resembling a "stout male human with... (other descriptors)", so it's apparent this is a fey spirit and not a race of people.

Four years later, Mythic Odyssey of Theros came out (2020). This book is a campaign setting, and here the satyrs join minotaurs and leonin as races- actual people with cultures, and of course, they keep their mythological sterotypes intact when possible (Theros is based on a setting in Magic: The Gathering). This is not the most sensical way to build a Greek world, basing it on a card game and maybe a few books (I assume they still make MtG books?), but if you read it, it's not like, garbage or anything- it's just not how I would run things.
This book gives us two more statblocks for satyrs to encounter:
https://drive.proton.me/urls/5FWPC25YN8#V37dPdgTYJYV
https://drive.proton.me/urls/7AWBD4EDFM#Aijjh2siS97e
In addition to the stuff from the monster manual, these satyr can't be put to sleep by magic, which is a fine perk, but we aren't really sure what interaction fey has with sleep and sleep magic. Do elves have this resistance because they don't sleep? The ability in 5e is called "fey ancestry", so do most or all fey never sleep? While an acceptable perk for an NPC, this isn't the main feature of a satyr.
This book also gave us the first of two playable satyr:
https://drive.proton.me/urls/GRF71SYXHW#L9hycHtAH3VC

The detalis in here are very good, and obviously suitable for Theros. Are they good enough for all satyr? They recieve extra proficiencies (above budget technically), but they are not in really great proficiencies like Perception or Stealth. They have a natural weapon that is generally not going to be something anyone can use (a satyr monk would have 1d6 attacks a couple levels early, a very minor perk). The extra speed has been changed to 35, which is about as fast as most PC races can have baseline (exceptions could be made of course, if something was all about being speedy I guess), matching the PHB's wood elf. And we have the real version of magic resistance, working on spells and magical abilities. This created a bit of a stir on forums at the time, as when a PC interacts with the world it's not always clear what counts as a "magical ability". Magical traps in the DMG all do, but there's no rule about traps in general, nor is the obvious guidance provided ("if you write a magical effect, describe it as magical so it interacts properly mechanically" is not advice given anywhere, despite being really clearly important). Dragon breath isn't a magical effect as written (this produces some other comedy interactions such as blasting the PCs in Leomund's Tiny Hut (Yfere Dome in this game), but should it be?

In 2022, they released a book that added some new races and "patched" a whole bunch of older ones. In certain places, such as adventurer's league, this means you couldn't play your older Satyr. The new one has almost no details and fits on one page:
https://drive.proton.me/urls/CE2R64RRMC#wwfkcbbAp8kJ

This version nerfs the magic resistance trait (now it's just spell resistance), keeps the mirthful leaps and 35 foot speed intact, and keeps the generic headbutt thing. The rest of it is left very generic.
Note that all races released after Tasha's have this generic ability modifier set, which is objectively a buff from a player perspective but is seriously lame.

The version we have is:
https://drive.proton.me/urls/HMMS069S18#nE1Rj4LWRG3z

Our emerald dragon narrator makes it clear that satyr are immortal fey spirits that incarnate at times and can have children with mortal women or simply instantiate out of a tree when they feel a desire to walk Nylsanna's Stone for a time. This means that while there's plenty of satyr they don't have some inherent physical need to establish settlements or control land and other such things, all of which I consider kinda contrary to the idea of being a satyr and wasn't what I wanted for Nylsanna's stone. The rules text I wrote makes it clear that satyr are male and that there aren't any female satyr.
I changed the horns to eventually count as magical at 6th level, a buff that shouldn't ever matter for an ability that is mostly there as a ribbon (the horns ARE a weapon, after all). Like all races I included, I changed the attribute pieces to be a hybridized so you can put a +2 in more than one place without having total freedom. Magic resistance I figured to include the older, unnerfed version- I decided that this was a main pull of the race, mechanically speaking, and it would be a very solid ability based on my assumption that things that counted as "magical effects" would be pretty common. For instance, the giant magical tunnel which tries to control your mind is a magical effect, so Geddreng had advantage on that save.

Also note that if I had gone with the "spell resistance only", then I think this race would be mechanically underbudget (especially compared to other races on Nylsanna's stone), and I wouldn't even know what to add to it that a player would expect or be happy with.

Anyway, the salient point is, I really didn't like the 2022 satyr nerf and I didn't seriously consider including it in this race.