Tracking Issue for the VSCode extension republish

[Veykril]

Tasks we can do right now:

• Verify the rust-lang publisher: Consider becoming a verified VS Code publisher #11978 (comment)

Tasks that have to happen somewhat simultaneously:

• Fix the config format: Config revamp #12010
• Get in touch with the vscode marketplace team to initiate the move

  Yes, we can move the extension. For that the author of the extension needs to send an email to vsmarketplace@microsoft.com and me inikolic@microsoft.com and we can do the rest.

• Fix up CI to publish under the correct publisher
• Fixup remaining links to the old marketplace extension

[rylev]

I believe another todo item here should be to update the tagline for the extension which still says "An alternative rust language server to the RLS".

[flodiebold]

See #12242