I wish them luck defeating that argument in court.
That is not the usual definition of archiving. That is just piracy.
Kind of a non-sequitur from you here.
Insofar as it maps to anything that has special status in copyright law distinct from making a copyright-violating unauthorized copy, it means making a copy of a computer program (not any other kind of work, such as an audiovisual recording) of which you own a physical copy to safeguard against the risk of destruction, media failure, etc., of the original copy.
("Personal archiving" is also commonly used to refer to making backup copies of one's own works and documentation that one owns that is not creative work subject to copyright, but that is less relevant here.)
That is considered time shifting which is not what downloading a YouTube video is.
No, they don't.
>He is downloading a video made by another user
Without a license from that user to make a copy of their video.