Thanks to @OwenM for pointing towards the solution in the comments (that were subsequently moved to chat). With that I was able to find an explanation in the book itself, in chapter 9, Figure 9-24, which "compares the B Mixolydian mode of E major with the seventh mode of C melodic minor".
This represents "chord=scale" thinking, where the player associates a chord with an entire seven-note scale, from which the chord is thought to "come from". In the example above, Levine equates the B7 chord with the B Mixolydian mode and takes the note spellings from the E major scale. And he equates the B7alt chord with the B altered scale, using spellings from C melodic minor.
Levine writes that "the eleventh can’t be lowered, it would then become the major third"... Yet in a sense he has indeed lowered the eleventh (E), which has become the major third of the chord.
So, in the A7alt chord, he has taken the note spellings from the Bb melodic minor scale.
He wants to see a Bb melodic minor ("jazz minor") scale on the keyboard with just the root moved to A, rather than an A something scale with lots of individual alterations. This avoids the problem of having (to write? or to think about?) both a flat 9th and a sharp 9th relative to A. Look at it as Bb melodic minor, and then every scale degree gets its own staff position. Whichever way you write it, it's always an approximation and favors one perspective over others.
... but other examples may follow a different logic
If you look at page 3, there's an "E7alt" chord where the same logic has NOT been applied. In the realization of that chord, Levine has written a G# instead of the Ab that he should have gotten from the F natural minor scale. This can mean either that (a) it was a mistake and he forgot to follow the logic, (b) or he imagined a different scale in that case. On page 80 there's some discussion about enharmonic spellings and different perspectives to 7alt chords, and as an additional view he mentions a tritone substitution, which again can change the spellings, if you think that way. The book explains that there are many ways to look at the chords and scales, and none of them is clearly the only correct way.
Is this a music theory book?
If music theory is considered as descriptions of the musical practices of some people, described by someone, for someone, then Levine's book is a theory book. Levine describes the practices of a certain group of people, using a certain style of describing. Both the practices and the description style may not be what you're used to. You have to take it like any human communication - subjective and context-dependent. Adjust accordingly. It must be said that Levine doesn't make understanding him too easy. This book should definitely come with a better how-to-read guide.
How to read the examples
From what I can see, there are two worlds, or coordinate systems in Levine's book. Or spaces? Traditional harmony space and keyboard space? The chord symbols come from the traditional harmony world, where the third of B7 is D#, etc. But on the piano keyboard, Levine imagines a scale that's based on C melodic minor, even so much that he takes the note spellings (even though no actual key signature is written) from there. Just the root note has been moved to the chord's root. The sounding note that in the traditional world is called D#, is played with the keyboard world's Eb note. In this chord=scale, but in some other chord=scale, it might be D# again, I guess. Or whatever scale Levine was seeing on the keyboard when writing the notation.
In this way of playing, every chord symbol represents an entire scale for the player. In more traditional harmony thinking, a chord is just its sounding notes, and for the rest, things are left open. But in Levine's "chord=scale=mode" world, every chord symbol means a mode. Which mode you imagine for a chord symbol, is subjective and up to tasted. And maybe that's why the enharmonic spellings can be different case by case - Levine was just imagining something slightly different that time.
Are there people who have read the book more thoroughly? Did I get this right?