The verb すわる "to sit down" is a punctual verb (瞬間動詞). The word punctual comes from the word point, as in a single point in time. These verbs have no duration—they take place in an instant, representing a transition from an old state to a new resultative state. In the case of すわる, the resulting state is being seated. (You can read more about these verbs on Taeko Tomioka's website and elsewhere).
Here's the problem: when you combine a punctual verb with 〜ながら, it's impossible to interpret it as "while" because the action has no duration. Instead, the only interpretation available is ながら's other meaning, which is a "counter to expectation" interpretation (similar to English "although").
To illustrate this, I'll borrow an example from Tense and Aspect in Modern Colloquial Japanese, p.108:
Although (he) graduated from college, (he) can't get a job.
Here, ながら attaches to the punctual verb 出る. Since this verb has no duration, it can't mean "while", leaving only the "counter to expectation" meaning. The expectation set up by 大学を出る is that the subject should be able to get a job, but in spite of that (as we see in 職に就けない) he cannot.
Unfortunately, neither interpretation makes sense in your example:
Since すわる is punctual, it has no duration and ながら here cannot mean "while". And there's no expectation set up by いすにすわる that is contrary to 音楽を聞いている, so the "counter to expectation" meaning doesn't make sense, either. For this reason, I've marked the sentence with a × and declined to translate it.
You don't have this problem with the simple connective 〜て:
The person who's sitting in a chair and listening to music is William.
Last, let's look at your example with 聞く. This verb is a durative verb (継続動詞), meaning that it has duration—it lasts longer than an instant. That means the "while" interpretation is available:
Henry is dancing while listening to music.
That's why this sentence works, but the other one doesn't. I think your book is right.