That's right, I remember the FASA explanation.
Evidently, the "trans" in transwarp refers to transporter technology, hence the "beaming of the warp envelope" idea.
Then there's the explanation in "Mr. Scott's Guide", which is a totally different theory.
I'll add one more to the mix, from the Spaceflight Chronology.
Written at the time of TMP, it postulated an imminent new breakthrough in warp technology (which was seen on-screen two films later in TSFS). The book called this breakthrough "super-warp" rather than the later "trans-warp". It all seems a variation on sonic speeds (supersonic, hypersonic, trans-sonic, etc)
In any case, the idea was that warp speeds would be bumped up exponentially. Instead of the warp factor being cubed times lightspeed, it would advance to the 4th power.
Warp 6 = 6^3 lightspeed (216 times lightspeed)
Transwarp 6 = 6^4 lightspeed (1296 lightspeed)
Any given superwarp/transwarp factor is to the 4th power rather than the 3rd power of warp factors.
Interesting idea, for whatever it's worth.