A company is running a modified version of a GPL'ed program on a web site. Does the GPL say they must release their modified sources?
The GPL permits anyone to make a modified version and use it without ever distributing it to others. What this company is doing is a special case of that. Therefore, the company does not have to release the modified sources.
It is essential for people to have the freedom to make modifications and use them privately, without ever publishing those modifications. However, putting the program on a server machine for the public to talk to is hardly “private” use, so it would be legitimate to require release of the source code in that special case. Developers who wish to address this might want to use the GNU Affero GPL for programs designed for network server use.
In case you missed it: However, putting the program on a server machine for the public to talk to is hardly “private” use, so it would be legitimate to require release of the source code in that special case.
There are numerous other potential consequences to the GPL and the question of when propagation occurs is ambiguous.
In fact, you are actually the one who is grossly misguided. Per that exact quote,
The GPL permits anyone to make a modified version and use it without ever distributing it to others. What this company is doing is a special case of that. Therefore, the company does not have to release the modified sources.
Specifically
Therefore, the company does not have to release the modified sources.
The faq is explicitly stating that the GPL would not require releasing modified source, and that if you want to force the releasing of modified sources, you should use the AGPL, as it has a clause to cover network server software
As your quote says, it would be a legitimate request... but not one made by the GPL. Instead, if you wanted to require this from your users, you would have to use the Affero GPL which was written for this very purpose (ie to close the "application server loophole" in the normal GPL).
No rate limiting, but it has pretty much everything else. If you want to add it I would welcome a pull request :)