Enter the matrix: the four types of Postgres extensions

Before working for a Postgres company, I had never used extensions.

Now, I'm part of a team working to fully automate turning on any extension. I didn't find great resources to explain the process of turning on an extension, and why it varies between different extensions.

I want to share my mental model for the different types of extensions, how to know what type of extension you're working with, and how to get it turned on.