PMS for Mac 1.28.0.5998
Plex for Mac 1.49.1.3146
Plex for Apple TV 8.5
Plex for iOS 8.5
Using Plex for Mac, I see in what appears to be an activity menu (sort of like ∿ ), that whenever someone is watching Live TV, the entry “Recording Live TV - Session” appears (in two lines):

How do I cause this Recording [of a] Live TV - Session to be saved as a recording that can re-played later.
If I find the show in the Live TV > Guide, then initiate recording, it only records from the point of initiation, it appears.
When my wife, using Plex for Apple TV, is watching an OTA Live TV show that is not scheduled to record, if I attempt to watch the show on Plex for Mac (so as to trying initiating recording), Plex for Mac hangs without ever displaying any show content—I then have to Force Quit Plex for Mac.