OST / LE / SE: First we had the Standard Jewel Case soundtrack for each movie. I tend to call these the OSTs, an acronym that means original soundtrack. But I'm not sure if that's exactly the right phrase. In tandem with these, limited (LE) and online limited (OLE) editions of what was essentially the same music (with one exception) was released at the same time. These did have different packaging but the only difference in audio content was the inclusion of bonus track on the TTT Limited Edition and a bonus DVD with the ROTK LE.

TRILOGY SET: These three CDs were also released as a trilogy set with some collectible cards. But again, the audio content was the same as the OSTs.

COMPLETE RECORDINGS: The concept for the Complete Recordings (CR) was to provide the soundtrack for the movies as heard in the extended edition CDs. But they didn't stick strictly to what is heard on the EEs. Some music was expanded so as to be longer than in the movies. Unused music (music taken out in the final edit) or alternate music (music scored for a scene but replaced by another piece for the final edit) was sometimes inserted in its proper place. In addition to these, the liner notes included short but nicely detailed notes about the themes in the soundtrack. And an annotated score (AS) (free downloadable pdf) was made available for each release that provided a running track by track analysis, a full accounting of the complete lyrics (see note that follows) and information on the artists and instruments heard in the soundtrack. The liner notes and annotated score were written by Doug Adams. There is a CR set for each movie.

(Complete lyrics are the full song/poem from which the lyrics as sung in the soundtrack were drawn. For the most part, one can not look at these and 'sing along' with any music in the soundtrack.)

THE MUSIC OF THE LOTR FILMS: This was slated to be a book with multimedia content.

The book would contain the information included in the CR liner notes and AS but would contain more information on those themes and more themes not mentioned in the liner notes. On his FAQ page, Doug writes:

The book will contain all the information from The Complete Recordings liners and the Annotated Scores (though somewhat restructured), plus several sections of text never before released.

The multimedia content would contain an assortment of things. From his FAQ page Doug had said:

We are planning for the book to come with 1 or 2 multimedia discs. These discs will contain analytical material as well as The Lord of the Rings Rarities Archives. The Archives will house the alternate music from LOTR for educational/study purposes. (magpie adds: otherwise... not set up for recreational listening). We're also looking in to including video material as well as secondary audio tracks with commentary.

Please note the emphasis on 'had said'. This is no longer the case. Keep reading.

The Music of the LOTR films would be available in two formats: standard & deluxe. The differences would be purely asthetic and Doug has been clear that there will be no difference in content.

LOTR: THE RARITIES ARCHIVE (fourth quarter 2009): Rather than the Rarities Archives (RA) being included as a section in the multimedia discs of Doug's book, they will be released on their own by Warner Brothers Records (WBR). What this means is two fold:

There will be substantially more rarities material released. Doug's project was limited in terms of budget (buying rights from Warner Brothers). WBR owns all this so licensing rights is not an issue. Doug wrote:

This means that CONSIDERABLY more music is now available for the project! You will see why this qualifies as something "major," I promise. And as I've said, there is some incredible LOTR music that no one has ever heard. Now we can be sure you'll hear it all. Frankly, it'll be amazing to put everything Shore wrote for this project side-by-side. It's remarkable both from a collector/completeist standpoint and from an artistic standpoint. Here, finally, is the complete collection of Howard Shore's music for The Lord of the Rings.

With the RA moving to a separate release, Doug's project now has room to do other things. What those 'things' are is still in development but Doug wrote:

We're still discovering the parameters of this new aspect, but we're using our public appearances and pre-concert lectures as a starting point—the experiences that Howard and I have had discussing this score. The experiences themselves aren't going to be the focus, obviously, the music will. But I think our framing device will develop along these lines. We're in an even more embryonic state on this than on the Master Box. It's come as a complete surprise… and wonderful surprise at that, but thoughts are still being organized.

The Rarities Archive will be available for purchase on its own.

MASTER BOX (official title pending):
WBR will make available a boxed set that will include:

CR-FOTR
CR-TTT
CR-ROTK
LOTR: RA

THE WHOLE ENCHILADA (official title pending):
WBR will make available a boxed set that will include

CR-FOTR
CR-TTT
CR-ROTK
LOTR: RA
The Music of the LOTR films by Doug Adams, deluxe edition

THE MUSIC OF THE LOTR FILMS:
Aside from a refocus from Rarities to other material for the multimedia discs, the release date has been pushed from fourth quarter 2008 to fourth quarter 2009. It will still be available in both standard and deluxe editions.

From what I've been following these past years, I'm still unsure I can definitively put a finger on it. There seem to a lot of things that comprise a "rarity". Generally speaking, what you said is correct - "music that was not used for the film but was considered". But there's more. Things that come to mind are:

• Alternate passages of music that were written for the same scene
• E.g., listen to the music for the lighting of the beacons on the ROTK:OST (2003 release) and the music for this scene on the ROTK:CR (2007 release). They're different because, as Doug says in the ROTK:Annotated Score, "the original version... heard on the OST... features a different harmonization of the Gondor theme—notably stressing major modes. The composition was reharmonized in the finished film to increase the dramatic tension and underscore Gondor’s dire situation.". Now neither of these are rarities of course because they have both been released. But there is muisc out there that fits this mould.
• E.g., listen to the music for the "Argonath scene", either on the FOTR:OST (2001 release) or the FOTR:CR (2005 release), and then listen to THIS podcast (jump ahead to 14:56) where you can sample a bit of music that was written for the same scene. It was a little teaser that Doug left us with after all the podcasts (6 in total) for the three CR releases were over and done with.
• E.g., Compare the music for the Breaking of the Fellowship that can be heard on the FOTR:OST (2001 release) and the FOTR:CR (2005 release) with the music that can be heard at the very end of the FOTR:EE Fan Club Credit scroll. The music is for the same scene but the former was performed by The London Philharmonic (if I recall rightly) while the latter by the NZ Symphony Orchestra (if I recall rightly) .
• Pieces of music that were written to different concepts of the film
• Shore worked in very close collaboration with PJ so that he was essentially writing music as the film progressed (and we're all aware just how many changes of concept the films went through). It's a little hard for me to elaborate any more since there exists only one example of this (that I know of). It is the prologue sequence. The music originally written for the initial edit of the movie sequence can be heard on the FOTR:OST (2001 release) and the music written for the film sequence as we see it can be heard on the FOTR:CR (2005 release).
• Drafts of music & synth mock-ups
• This represents concepts of music that were written but were not worked through all the way to a finished piece. From my understanding, synth mock-ups is the process of using a synthesizer to replicate vowel sounds in order to listen to what a choral piece would sound like if it were sung by a real person. See here for a little discussion that happened some time ago concerning this.
• Songs that were NOT written for movie sequences
• Two things come under this category. One is likely to be something like a synth mock-up but with actual lyrics (even the lyrics were reworked so that the songs you hear in the final versions of the films are not necessarily how they were initially written) and complete songs that were written and recorded but don't fit the narrative of the movies per se. Two examples of this spring to mind.
• Bilbo's Song - It was written by Shore "because he felt like" after he finished writing the music for ROTK. It wasn't written for any movie scene and it can be heard as the ROTK Fan Club Credits scroll on the ROTK:EE. This song however is no longer a rarity because it has been released as the last track on the ROTK:CR (2007 release).
• Use Well The Days - This was a song co-written by Annie Lennox before Into The West was written. It was released on the DVD that accompanied the ROTK:LE (Limited Edition) that was released in 2003. It is still considered a "rarity" since it was not really released as part of the standard ROTK:OST not the ROTK:CR. And Doug has said that it will be there on the rarities disc(s).

You might've heard smatterings of some rarities if you've been keenly listening to the documentaries on the DVDs, the music that plays over the DVD chapters etc. The most notable of these to my mind are:

• In THIS podcast same as above but jump ahead to 12:44 and listen till 13:55. You can hear a little bit of alternate Shire-music under Doug's commentary about what the rarities will be.
• The unused Flight to the Ford music that I pointed to in answer to question 1 above.