All I need is to keep track who joins, updates, and deletes their membership and which specific tier

The pledge related ones now have members: in front of them, other than that they are the same triggers:

https://docs.patreon.com/#triggers

My initial expectation is that “create” is when someone joined a membership, “update” is when they switched to a different tier or probably also updated their own user information, and “delete” when they leave the membership

members:pledge related triggers should be enough for your use case.