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.

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!