I.e. - if a TM accepts the word 0011100 on one path, while on another the same word is rejected. Do we say that 0011100 is part of the language, or do we say that it is not a valid TM?
A non-deterministic Turing machine accepts a word if an accepting computation exists. There can be many non-accepting computation paths. This is what makes recognizing the complement of the language more difficult than just inverting accepting an rejecting states as you would do with a deterministic machine.