It’s just a tricky one to get defined right. I would at the very least expect a recording of a sound check, where a performance is identical to that at the later concert other than a lack of audience, to be as “live” as a recording of that concert.
Also, define “audience”. If there’s 10 people (other than performers/engineers/…) in the studio while the recording happens, does that make it live? If not, then a concert recording where only 10 people show up would also not be live, would it.
So at its base, what I think is always going to be required for a live recording:
- a) all parts performed and recorded at the same time (I’d add mixed, but many live albums have mixing applied later), or
b) the “main” performance(s) (typically vocals, but could also be a solo trumpetist etc) are live, but there is a prerecorded backing track.
- no “postproduction” other than mixing/mastering (no adding extra tracks, no “polishing up” of vocals, …)
The presence of an audience (whether physical or via broadcast/streaming) to me would not matter much for 1a, but would strengthen the 1b case.