"a soundtrack still implies that it is songs from the show."
Ummm..no. The term "soundtrack" does not mean that at all.
It actually means the audio strip along the side of a film. In the 1940s M-g-M records pioneered the idea of recording selections "directly from the soundtrack" and issueing them on records. These were called Soundtrack Albums. The industry later abbreviated it to just the word "Soundtrack."
Stage shows do not use pre-recorded audio tracks. The casts perform live every night. The recordings we buy are studio recordings made with the home listener in mind so much of the material has to be changed. It's almost like doing a radio version of the show, condensed to about one-hour. Cast albums are a curious amalgam. They never record exactly what is heard on stage but they create that iullusion. That is why so many flop shows left behind great sounding cast albums.
While it IS correct to call movie sountracks "cast recordings" (since they are the movie casts), it is incorrect to call a cast recording a soundtrack.
For GREASE you can buy the Broadway cast, the movie cast, or the Broadway revival cast. For SOUND OF MUSIC you choose from the Broadway cast, the movie cast, the Broadway revival cast or the London cast.
In short, all we have to do is lose the word soundtrack and refer to all show recordings as cast recordings. CR for short! The London CR, the movie CR, The Broadway CR, The Timbuktiu CR....
Cast albums are NOT "soundtracks."
Live theatre does not use a "soundtrack." If it did, it wouldn't be live theatre!
I host a weekly one-hour radio program featuring cast album selections as well as songs by cabaret, jazz and theatre artists. The program, FRONT ROW CENTRE is heard Sundays 9 to 10 am and also Saturdays from 8 to 9 am (eastern times) on www.proudfm.com