I went in the chatroom and said a rarity, Then I went in the game and the rarity was different from the chat's rarity. Why did it do this? Late update?

If you searched up a Pokemon still obtainable some more might've been caught/bought by other users in the time frame between your search.

I don't think there should be a discrepancy, because Creed Bot is Creed Bot, the artificial "intelligence" of pokemon creed (lol, I had to taunt the Bot a bit) who manages it all, and it's the same as the total amount of money in creed, he counts all the pokemon and updates the rarity list based on that. (Not literally, it's all automated, but I'd hope that helps you understand.)