"Time shifting" started in the mid 1970's with the introduction of VCRs. It simply means watching a TV program at a different time than it was originally broadcast.
Before VCRs there was no "time shifting" of TV at home. If you wanted to watch a show, you had to be in front of your TV when the network decided to show it. And if you need to go to the bathroom, tough luck, try to hold it in until there is a commercial.
With VCRs you had to wait until the program was finished before you could watch it, but with the introduction of DVRs in the late 1990's, it because possible to start watching programs while they were being broadcast. (When you use pause & rewind on a "live" program, the DVR is recording the program; the DVR just isn't programed to save the recording once the program is over.)
The term has also been used with audio & internet programing.
>why they chose that term when everyone was more than likely going to think "time travel"
It made perfect sense when VCRs first came out, and even after "time shifting" TV for 35 years I'm still not confused.
>Pretty sure it doesn't have anything to do with time travel
It's moving a video stream into the future. How is that not "time shifting"?