So let's look at some of the existing ones like Wikimedia or Apache. What are they doing to claim that particular exemption?

Here's Wikimedia's confirmation of exemption letter.

https://upload.wikimedia.org/wikipedia/foundation/9/90/501%2...

To go further, I'd look up exactly what "Public Charity Status 170(b)(1)(A)(vi)" is, and then compare that to how they describe themselves here: https://wikimediafoundation.org/wiki/FAQ/en

In a different thread a couple of days ago, someone mentioned that Apache got a scientific exemption.

Which is not surprising at all, since the Apache HTTP Server had its roots in the National Center for Supercomputing Applications (NCSA), a scientific institution.

Python Software Foundation also got a scientific and educational exemption.

I expect that they aren't claiming a scientific purpose. More likely they claim to be educational, or that they provide a public work.

at least in the case of wikimedia, software is merely a byproduct of their goal. To quote their main page:

    Wikimedia is a global movement whose mission is to bring free educational content to the world.