Pretty sure it was this one: https://en.wikipedia.org/wiki/Sony_Corp._of_America_v._Universal_City_Studios,_Inc. Sony were actually the defendant, with their Betamax format. It does seem to focus primarily on time-shifting, ie recording live to watch later, however the reason for this was that the content was already available to the viewer and thus the copying should be permitted fair use. The Supreme Court also quoted Mr Rogers’ testimony in their ruling.

“Some public stations, as well as commercial stations, program the ‘Neighborhood’ at hours when some children cannot use it. I think that it’s a real service to families to be able to record such programs and show them at appropriate times. I have always felt that, with the advent of all of this new technology that allows people to tape the ‘Neighborhood’ off the air, and I’m speaking for the ‘Neighborhood’ because that’s what I produce, that they then become much more active in the programming of their family’s television life. Very frankly, I am opposed to people being programmed by others. My whole approach in broadcasting has always been ‘You are an important person just the way you are. You can make healthy decisions.’ Maybe I’m going on too long, but I just feel that anything that allows a person to be more active in the control of his or her life, in a healthy way, is important.”

Applying this reasoning to new technologies has since been debated back and forth through the decades with little clear resolution. Subsequent cases have sided with the rightsholders (eg against Grokster and Limewire), but the reasoning behind them was all over the place. They addressed the purpose of file sharing technology and concluded that those services existed primarily to facilitate copyright infringement, rather than addressing the matter of personal backups.