Many improvisers, non-improvisers and anti improvisers alike continue to question, in print, the recording of improvised music. There have only been a few exceptions who have appreciated the intrinsic necessity of recording improvised music, such as Jean Dubuffet ("It is impossible to write true music, except with a stylus on the wax, and this is what they do now in recordings") and Leo Smith ("In our times now, an oral-electronic tradition is being born, and this signifies the age of a new improvisation-art-music-form"). Fortunately, most improvisers have ignored their doubts expressed in print, and have gone ahead and recorded their music.
In the past, great improvisers (the likes of Bach, Beethoven and Chopin) had to resort to notation as the only way to preserve their music. Those that did not do so have been reduced to a footnote in someone's memoirs (if they are that lucky). The advent of recorded sound has meant that great improvisers can now preserve their music directly without having to resort to the vagaries of notation. The advent of recorded sound has also meant that the notation of music has become somewhat superfluous. (The advent of the player-piano and electronic sound sources has also reduced the need of would-be composers to use notation and treat musicians as fodder.)
Thus, recordings and improvisation are entirely symbiotic, as if they were invented for each other (although Edison, among others, does not seem to have appreciated this). Also, since pre-composed music is preserved (albeit imperfectly) as a score, it is much more important to record improvised music (which is otherwise unpreserved) than pre-composed music. (It does seem particularly pointless to continually re-record the same pieces of pre-composed music, especially when there is so much music waiting to be made, and when most of the published performances are mediocre or worse.)
Objection has been raised that a sound recording cannot contain many of the factors that influence improvised music, such as the acoustics, the audience (if any), the relationships (if any) between the performers, the weather, the economic climate, the contents of breakfast (if any), etc., etc. This is entirely true; but it is doubly true of pre-composed music, since similar sets of factors influence both the composition and the performance of the music. Then again, members of an audience at a concert normally do not know all the factors influencing the music being performed, so they are not much better off in that respect than listeners to recordings. (This could be an argument against both the performance and recording of music, in line with the CANDID CAMERA silent pianist who claimed that the sound of the music distracted from the music itself!)
The problem of what to publish on recordings then arises. Should it be representative or just the best? Should it be as realistic as possible, or the creation of something that cannot exist outside of a sound recording? Ideally, every music performance should be recorded and the recording published complete. This would give the listener a true perspective (especially if a list of the prevailing extra-musical influencing factors were also included). Unfortunately (fortunately?) no-one has the resources (energy and/or multi-processing capability) or freedom (time and/or money) to deal with all this.
Hard and fast rules are impossible. Reality is generally preferred by this listener, but a few "artificial" creations (usually involving overdubbing) have been enjoyed. Complete performances are more satisfying theoretically, since they get the listener closer to being at the concert; but many performances unfortunately contain both good and bad bits. This record buyer begrudges paying for the bad bits, and does not have the time and desire to sit through them repeatedly (unless they are bad enough to be perversely enjoyable).
The choice of material to be published is also restricted by the form of the media, i.e., two lots of about three minutes on 78s, two lots of about twenty minutes on LPs, one lot of up to 80 minutes on CDs, etc. The recording editor, normally (hopefully) the musician(s) involved, thus becomes a composer who has to fill certain allotments of time with amounts of pre-recorded music, that may or may not have started out as improvised. But then, the act of improvisation itself is filling time (either a predetermined or open-ended amount) with music - something that could be called real-time composition, and something that has more need and more right to be recorded than anything else.
1997 afterthought: The advent of DVD with its eight hour capacity somewhat mind-bogglingly changes ones thoughts of how to utilise the media!