My understanding of the extension situation is in some ways similar to HanonO’s, and in some ways different.
My impression of the extensions submitted to the public library (similar to what HanonO said) is that, since these are the official extensions that story authors can download through the Inform IDE, they have the strictest requirements, to make sure they work and are easily usable by story authors. These extensions must be submitted by email to the Inform librarian, who will then check them to make sure they follow the requirements.
The ones in this repo on Github github.com/i7/extensions are described like this: “some may be ready for public use, others may be barely working experiments.” In practice, I see this github repo being used for extensions that are works in progress, that multiple people are collaborating on, that are broken and in the process of being fixed, or that the extension author has no intention of putting in the public library (e.g. because the extension author doesn’t want to have to maintain it, or doesn’t want to update it for the latest version of Inform, or because the extension author abandoned the idea for a better one). Anyone with access to this repo can upload their extensions, anytime. So it’s “unofficial,” and more informal than the public library.
At one point, my understanding was that the extensions on the Inform website at inform7.com/write/extensions/ are older versions of public library extensions, for older versions of Inform. But as I look at it right now, I am seeing at least one newish extension on there (but not others), so I don’t know whether something has changed, or if I have just misunderstood the purpose of this section of the site.
Then there’s a fourth place for extensions–the Inform 7 extensions library site which Dannii had put together–that was intended to be a solution to the issue of I7 extensions being spread out in different places. However, some details about how the site was supposed to function–mainly, how one was supposed to differentiate between extensions that were in different states of completion/function–were never totally clear to me. Here’s the github repo for the site: i7el.herokuapp.com/