Let's go a little in a different direction and talk about remasters, remixes, alternate takes, and definitive editions.
When there's a crappy old release, and there's a desire to clean it up, there are a lot of ways you can do it:
A purist would insist on simply improving the transfer between the 'master tape' (traditionally some high-quality reel-to-reel tape) and the final product. Companies like MFSL will do this at half speed, for instance, to reduce hiss (if you do it at half speed, the hiss sounds the same, and then when you play it at full speed, the hiss is much higher, and much of it is out of the range of human hearing).
If you want to get less pure, you can apply all sorts of equalization and similar things. One can argue that no matter what you do, your transfer will never be 'pure', and that you need to compensate in these ways.
You can go further, and apply noise reduction (something I personally dislike, as I can often hear its effects on the sounds themselves more than on the noise), gating, things like that.
You can go even further, and take as your source material the original multitrack tapes, rather than the master. This means you can apply any one of the above steps to the original, cleanly recorded instruments. The big problem is that you have to actually duplicate the parameters of the original mix as closely as possible. The end result is likely to be less 'definitive' in that it sounds less like the original, but who's to say if it's less 'definitive' in the sense of closer to the original intent?
You can go a little further, even, and intentionally diverge from the original mix - a true remix. Again, this is a highly subjective matter.
Some interesting examples:
The Who - Who's Next. The original master tape was apparently ceremoniously destroyed after completion, so the remastered CD release was actually remixed. Some people complained of this. The original CD release was from some master other than the original, so is truer in some ways to the original recording.
Judas Priest - the remasters. Actually, these are again remixes, and some people have found some flaws, compared with the originals. Still, the sound quality is arguably better.
Judas Priest - Hero Hero. This deserves its own post (maybe, maybe later). Apparently no one was satisfied with the results of Judas Priest's first album's mastering (Rocka Rolla, 1974), so to capitalize on their success, their original record company (Gull Records), asked Rodger Bain (producer of Rocka Rolla, as well as early Budgie and Black Sabbath), to truly remix the album, as Hero Hero, in 1981. The results are interesting, where most of the songs are just 'interestingly different' (in my opinion), while a few sound actually better for the remixing. Certainly, you can hear Rob Halfords climactic "I can't go on" more in the front of the mix, though with a lot of echo, on "Run of the mill".
(for more info, check out the excellent Judas Priest INFO PAGES)
Various MFSL releases - Jethro Tull, Pink Floyd, all sorts. These are true remasters.
Apparently, the King Crimson remasters were really remixes, that angered some people..
For the rerelease of Koenhihyakkei's first album, drummer Tatsuya Yoshida not only remixed it, but rerecorded the drums.
Jethro Tull's Stand Up has had many releases on CD. The "Orginals" rerelease, by Chrysalis, was a processed remaster, with all sorts of noise reduction, compression, eq, and other processing, while the MFSL gold disc version had none of those features. The results are subjective, and also will sound different on different systems.
Given all these changes, it's sometimes difficult to even define what a 'definitive' edition really is.