Does anyone know how exactly the Growth Type works?

Have any specific ages for each type that reach the peak and start dropping? Almost all players are "Standard/lasting" here.

It should be Standard for those who kept the same form or didn't improve that much until their retirement if they retired or had their peak between 30 and 35 years (something like that). If that happened when they were around 40, then it's Standard/Lasting.
It should be Early/Peak for a player who had his peak at an early age like Alexandre Pato or retired before 30 years while you can give Early/Lasting to young players who weren't too good yet and improved year by year.
Late/Peak and Late/Lasting are like Early/Peak and Early/Lasting but refer to older players. If hey had their peak at around 30 years or more and retired at around 35 years then I give them Late/Peak, if they retired at older age like Dino Zoff it's Late/Lasting.

I did an experiment a while ago, testing all growth types on an unedited Other Leagues C player (age 20, all stats are 60), and here's the results to give an idea on how growth types work:
1. Early Peak: reaches his peak quickly but drops his form just as quick, peaks at 23 and stats dropping at 25, real life examples: Pato and Bojan.

2. Early/Lasting: peaks at 23 as well but lasts longer, he'll start dropping before 30 (at 27), real life examples: Gareth Bale.

3. Standard: as the name suggests, it's the standard for player's growth, usually it peaks at 28/29 and starts dropping instatly, this is the standrad for most players in the game and the created players as well, however for Other Leagues C players you'll get weird a growth graph, their form will drop at 19 and 20 and starts growing back at 21 until 25 where it'll start dropping, you could change that if you change the growth type for that player and then change it back to standard, then the graph peaks at 26 and drops at 28 which is much better than the old one.

4. Standard/Lasting: slightly longer peak and slower dropping rate, peaks at 26 and drops at 30 and if the player keeps a high match rating, he can maintain his form for a few more years, real life examples: Messi and C. Ronaldo.

5. Late Peak: slower growth in their 20, won't peak until 30 and lasts until 32, mostly used for goal keepers, it's also suitable for players who still at the top of their game in their early 30's like Lewandowski between 2019 and 2021 or those who reached their peak at a late age like Jamie Vardy or Emiliano Martinez.

6. Late/Lasting: very slow growth, peaks at 30 but lasts until 40, only for players who are still preforming great in their mid and late 30's, real life examples: Paulo Maldini and Javier Zanetti.

Keep in mind that peak and drop ages might change at bit depending on the player's assigned age and stats.