Some clarification on Lost Presents. You are limited to 2 presents exactly every 10 minutes. There is a desync issue where people see different presents from the majority of players that occurs sometimes and this is what causes the 3rd present to appear. Presents are also linked together across all 5 regions. So if you pick up Present #1 in Kanto the corresponding present will be gone from all 4 other regions.
Lastly regarding Big Presents they occur globally for every single player. Like Phenos it's still random if you will receive a Big Present or an Alpha Amoonguss / Electrode but every single player will have one occur at that location. These Big Present spawns are also linked across all 5 regions. So if one spawns in as a big present in Kanto there will be a corresponding Big Present across every region. They are LINKED though so you can't just go to the other region and get a 2nd Big Present that spawn.
As a side note when you unlock Big Presents Santa also starts rewarding you Mysterious Ultra Balls at a 100% rate for normal lost presents if you'd like to add that.