And now for the unofficial, fanfic version of the story...(really meaning here's my version of the story)
Transwarp isn't a name for any specific technology for faster than light travel. Rather, it's a general name used to describe any advanced warp technology relative to the standards of the day. For example, in Star Trek III, the U.S.S Excelsior had a Transwarp drive since the drive system (Koeller UTI K27) was faster and more efficient than the ones used by the Enterprise and other ships of the time. By the time of TNG, the engines of the Excelsior were no longer considered to be transwarp, since most every ship class was already using them, or their variants. During TNG, things like 'Quantum Slipstream', 'Soliton Wave Drive' and 'Transwarp Conduits' could be considered 'Transwarp'.