Looks like it means all sorts of things. Does anyone know what Tom Lord-Alge found it to be?
I haven't been able (yet) to find any first-person description of "flying vocals" from Alge himself, if that's what your looking for.
The description in the very Wikiwacky article that Svedde first cited at the top of this post put it this way:
Lord-Alge helped pioneer a method of digital tape editing known as "flying" vocals (e.g., using a piece of the chorus to accent a different part of the song, etc.).
Now, we all know that when it comes to this kind of drill-down detail, that one has to be careful of taking Wikiwacky at full face value, but what I find interesting is the author's combination of a specific meaning (somewhat different than what's described anywhere here yet), combined with the use of the term "etc."
I think today's general "definition" of flying vocals has been pretty much genericized to mean almost any repeated use of a vocal sample within a song. Whether it was originally specifically referring to the layering of vocals as Tom describes or something like the Wiki description, or something else, I don't know.
But I tend to look at it like this (rightly or wrongly); that Alge didn't limit himself to one particular application of flying in vocals. in fact, - as Tom also rightly mentions - it's surely not a technique that Alge "invented'; It's something that I'm sure has been done by countless others at one time or another since the beginning of magnetic recording.
Rather, where Alge's name becomes attached to it is in the idea of common or liberal use of it overall (once the ease of digital editing came around) as a distinctive mixing "style"...or maybe not style so much as a signature technique.
It's like crediting George Martin with the use of tape splicing or Phil "Psycho Killer" Spector with the layering of instruments. Neither one of them invented the basic technique behind the sound they became famous for, nor did they limit themselves to applying it in one specific manner, necessarily. But they became famous for their respective techniques because of their liberal application of it as a signature technique yielding a signature sound.
Just a viewpoint here; not solid revisionist history. If someone knows more or better, I'll take it!

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G.