This is a guess since I'm not a lawyer, but since users license their content to Twitter when posting it, Bright Data might have to prove fair use. I don't think that question has been answered yet in relation to AI model training, but search engines have been doing this for decades for what it's worth, so I don't know.
By attempting to exclude Bright Data from accessing public X posts owned by X users, X also nearly "obliterated" the "fair use" provision of the Copyright Act, "flouting" Congress' intent in passing the law, Alsup wrote.
Do they give up the copyright, or license it to the website? They still created the content, and I don't have a Twitter account, but after briefly reading the ToS, it says they license it to Twitter (which is pretty standard from the other services I use that I've read the ToS for).
The copyright law says that Musk cannot claim copyright while also claiming not to be accountable. That has nothing to do with users waiving their copyright, which has been the standard practice for all forum software since practically forever. It's why prior to GDPR, their was nothing to motivate websites from deleting your posts and even that isn't about copyright, it's about privacy.
Copyright can only be waived in the US by dedicating the work to the Public Domain. In most other countries, it can only be assigned or licensed to someone.
The "standard practice in all forum software since practically forever", has been to include a very broad use license on the work, without switching the copyright holder, in order to protect the forum owner from liability.
The GDPR is about a very broad take on "privacy", where the rights of "access, modification, and removal" get extended to any "personal information", no matter whether it's "personally identifiable" or not.