I have started reading "Jazz Theory" by Mark Levine. Note that I am a beginner so please be indulgent :)

At the beginning of chapter 2 he gives examples of II-V-I chord progression. However when I read those chords I don't understand their structures.

(note: now that I finished writing this post, I understand them better I think, however any insights is still very much welcome !)

First example: In the key of Eb:

Because the key is Eb then the II chord is a F-7, the V a Bb7 and the I a EbMaj7, this is clear.

First chord:

F-7 has the following four notes: F, Ab, C, Eb.
On the picture I do not see those four notes but instead: F, Ab, Eb, G.
From what I know it is possible to play a four-notes chord with only three voices, in that case we discard the fifth degree, thus the C. So ok i guess this is why we do not have a C in the chord.

Then what about the G? Is that because the G is the melody, so we have the F-7 chord on three notes: F ,Ab ,Eb. In addition there is the note G that is not part of the chord but just the melody.

Next chord: Bb7

In theory: Bb7= Bb, D, F, Ab
On the picture we have the notes: Bb, Ab, C, D, G
My guess: The chord is made of: Bb, Ab, D (omiting the fifth). The rest C,G is the melody?

Next chord: EbMaj7

In theory: Eb, G, Bb, D
On the picture we have the notes: Eb, F, G, Bb
My guess: The chord is made of: Eb, G, Bb. But then it would be a 7 chord without a 7 note?! The F is the melody? This one I do not understand....

One thing I noticed in this first chord progression is that the notes that I guess are part of the melody are all second degree of the chord (first chord is a F, the melody is a G; 2nd chord a Eb, melody is a C and its dominant; 3rd chord a Eb, melody is a F. Is there something to understand from this?).

Second example: in the key of D

Chords are E-7, A7, Dmaj7

First chord: E-7

In theory: E, G, B, D
On the picture we have the notes: E, D, A and a delayed G
My guess: The chord is made of: E, G, D (omiting the fifth) with A being the melody.

Next chord: A7

In theory: A, C#, E, G
On the picture we have the notes: A, G, C#, D#, E
My guess: The chord is made of: A, G, C#, E and D# is the melody.

Next chord: Dmaj7

The 4 notes are part of the chord.

Melody is made of the fourth notes of each chord (ok not the third one...) (chord E-7, melody is A; chord A7, melody is D#)

Third example showing two II-V-I chord progressions

One in the key of G, the second in the key of B

First progression in the key of G

First chord: A-7

In theory: A, C, E, G
On the picture we have the notes: A, G, C, E, B
My guess: The chord is made of: A, G, C, E and B is the melody.

Next chord: D7

In theory: D, F#, A, C
On the picture we have the notes: D, F#, C, E, A
My guess: The chord is made of: D, F#, A, C and E is the melody.

Next chord: Gmaj7

In theory: G, B, D, F#
On the picture we have the notes: G, F#, A, B, D
My guess: The chord is made of: G, F#, B, D and A is the melody.

Once again the melody is the second degree of the chord (chord is A, melody note is B, etc...). Just a choice or is there a logic that appears for a trained musician? Could it mean that from a melody B, E, A, I decide to harmonize this using a II-V-I chord progression with the melody note being the second in the chord (this probably makes no sense at all....)

Second progression in the key of B

First chord: C#-7

In theory: C#, E, G, B
On the picture we have the notes: C#, B, E, G#, D#
My guess: The chord is made of: C#, B, E (omiting the fifth) and G# and D# is the melody.

Next chord: F#7

In theory: F#, A#, C#, E
On the picture we have the notes: F#, A#, E, G, C#
My guess: The chord is made of: F#, A#, E, C# and G is the melody.

Next chord: BMaj7

In theory: B, D#, F#, A#
On the picture we have the notes: B, A#, C#, D#, F#
My guess: The chord is made of: B, A#, D#, F# and C# is the melody.

I would explain how the spelled-out notes and the chord symbols above them go together like so: In a song that might be notated with simple chord symbols like these ones here (e.g. Em7, A7, Dmaj7), you could play, for example, something like these notes, and it will accomplish essentially the same harmonic functions as if the written chord symbols had been played in their simple forms exactly as written.

Yes, he left out a fifth here, added a ninth there, etc. but he tried to do it so that the result is more or less compatible with the simple version. From the last chord of the first example he left out the major seventh. Maybe he didn't like it. To me, leaving out the maj7 sounds fine and doesn't ruin the tune, and ending the line to a maj7 would have sounded too kitchy. A soloist can look at the chord symbol and assume that it's safe to play a maj7 there, and that assumption is safe even if the pianist doesn't play that note (as long as he doesn't play a minor seventh or something, but that would be such a drastic change that any soloist should hear what happened and take proper action i.e. change the solo and/or throw something at the pianist). What do you think? Does it sound similar enough, so the tune is still recognizeable? Will a random soloist be able to improvise a solo there without your jazzed-up chords catching her by surprise and clashing with what she expected to be safe notes to play? Will you get punched by some drunk redneck because you made his favorite song too jazzy? Choices!

There is any number of substitution and modification tricks, and they all have slightly (or very) different tastes and levels of "compatibility". It's like cooking. You might use a different type of cheese in your pizza, and it will still work as pizza cheese, just a bit different. As you take progressive steps further away from the simplest Emmental or what have you, at some point you might cross a line where most of your dinner guests do not identify the substance as cheese at all anymore. Some of them might like it though, whatever it's called or wherever it came from.

How do you like this interpretation, does it still "work" in your opinion? What practical implications might this choice of notes have?

At least it sounds fuller? Too full? To me it feels intriguing that such a lump of notes makes any harmonic sense at all. If you're the soloist, you might do something like that as an experiment, but as an accompaniment it narrows down harmonic options too much. A soloist cannot e.g. outline an Abdim/Bb chord on the dominant Bb chord, because the piano's C note would sound nasty together with the Ab dim chord's natural B.

Probably Mark Levine's book tries to make you see the whole thing, and you're supposed hear a melody line there. To me the topmost notes clearly seem to be a melody. You can't learn harmony completely independently of a leading melody, because the melody notes are a part of the sounding whole anyway, whether you like it or not. If at first there's silence, and then you play or sing a note, the sound you made affects all musical dimensions: melody, harmony, rhythm, timbre, ambience, everything. We have got quite skilled at dancing about architecture, but music should be all about listening. Pure stand-alone harmony only exists in theory books.

In my opinion, if you play jazz, you're supposed to be able to take a simple song and "jazz it up" with various substitutions, omissions, voicings, rhythms. I'd call it improvised on-the-fly arranging. You do it very quickly, and should be able to improvise different notes every time. You cannot really learn to do it as a theoretical exercise, reading and talking about things in theoretical terms. You have to learn it by doing it.

What you should do with the book and the examples is, take other songs (preferably from outside the book) that have chords and chord progressions that you've learned substitutions and voicings for, and apply the tricks to the songs. Or just take any simple song and jazz it up in different ways. Imagine that you're playing a gig, and it's not a drill. Learn to understand how each trick affects your feelings and possibilities for alternative melody/solo lines. Try more and more and more melodies and songs... at some point you should encounter things that don't work that well, because e.g. your substitution trick added a note that clashed with the melody or an important bass line or something. And you do it the other way around as well, by playing different melody lines, i.e. solos, over existing chords. And you play single-note lines over just a bass line or even a drum beat (or completely solo without any accompaniment), and you notice that your single-note lines seem to imply chords in your mind, even though you're only playing one note at a time! And that's how you begin to really understand jazz harmony.

A good jazz soloist, in my opinion, imagines her own chords and outlines them with her solo. Improvised, on the fly, different chords each time if she so chooses, but it's the soloist's decision, not the comping guitarist or pianist's. That's why a good jazz accompaniment doesn't play too many notes. You leave room for the soloist to play different chords and scales. You should learn this from both the accompanist's and soloist's perspective.

There is a lot to go through here, but you seem to be on the right track. The author is adding the 9th to most of these chords, which is a fairly common thing in jazz, but it would be nice if he explained that. Sometimes it is because the note is in the melody, but not always (like the Eb major 7, in the first example).

Personally, I think is bad notation for an instructional book, and what you are seeing it is real a flaw. The Levine book is pretty good, but it is not great for beginners becomes he assumes you already know this stuff.

Other authors might label the chords in the first example as: F-9 | Bb13 | Ebmaj9, which would probably be more clear and cause less confusion.

Admittedly, I hadn't read all of your question, but the moment Levine was mentioned, I guessed what was coming. As I was reading the book, some time ago, I started making notes of 'errors'. There are plenty. And since it's supposed to be an educational tome, it didn't convince me. Saying one thing, then writing dots which don't agree, to me isn't particularly educational. Confusing is a more apposite word !

However, the basic ideas are all sound, it's just that Levine obviously likes to spice stuff up, but doesn't mention why. Which is a great pity.

For a more easily digestible and accurate read, try Bert Ligon.

To proffer some kind of answer to your many observations, Often, when playing jazz, the written chord gets changed a little (or a lot). It's when you see a X7 that you start thinking 'a 9 will fit over that, or maybe I can make it 7b5', so you end up playing a 'version' (not inversion!) of what's written.

This is what's been happening, so please use large pinches of salt when trying to unravel what's been written. The fact that you have found discrepancies shows that you are at least understanding what should be going on. Well done!