"The Annotated Turing" on listing all binary numbers between 0 and 1
You haven't shown the full proof, but it most likely has the form &quot;Assume that the set is countable.  Consider any enumeration.  Then (blah-blah-blah) shows that there is a number not in the enumeration.  That's a contradiction, so our assumption must have been wrong.&quot;
If that's the structure of the proof, then pointing out that the proof appears to say two contradictory things is not a surprise; that's entirely the point of the line of reasoning, to show that if you start with a particular assumption, you reach absurd results.
What's actually true?  The truth: that list doesn't actually contain all the real numbers.  Assuming that it does is a faulty assumption.
It's possible that the text could have been written a little more carefully to caveat that the statement about it containing all numbers; but oh well, no one's perfect.  Books written for a popular audience, in particular, aren't always the best source if you want a precise and careful proof of a mathematical theorem.