Definition of a prefix-free Turing machine
If you were using a self-delimiting machine as described, you have to change your notion of what it means for the machine to accept a string. You want to say that the machine accepts a string $\omega$ if it halts in an accept state with the read head on the last symbol. The idea is to think of the computation as a loop: it first asks whether $\omega\upharpoonright 1$ is in the language, then whether $\omega \upharpoonright 2$ is, and so forth, and it halts if it finds a prefix of $\omega$ in the language or if it runs out of bits of $\omega$. 

The point is that if the input tape contains an infinite juxtaposed sequence of nonempty words from the language, the prefix free machine can be used to parse out the individual words from the sequence.  The definition of a prefix-free machine from the book is designed for this sort of application, not for the problem of deciding whether an individual word is in the language.