Can a Turing Machine be defined over many languages?
A Turing machine isn't defined over a language.

Given a Turing machine $M$, we can define the language of inputs on which $M$ halts; or the language of inputs which $M$ accepts; or the language of inputs which $M$ either accepts or doesn't halt on; and so on.

Usually we are interested in either the language of all inputs on which $M$ halts, or (for a machine halting on all inputs), in the language of all inputs which $M$ accepts.