Do the time and space complexities of a language depend on the Turing machine which is its recognizer?
Yes, you are correct. I applaud your careful reading. 

This reading is most obvious if we view a TM as an algorithm (which they are). Clearly, for any computational problem, some algorithms are more efficient than others. Compare bubble sort to merge sort, for example.

But their definition is not wrong, just perhaps a little misleading. Any algorithm can provide a time-bound on a decision problem; but only one of them will be the lowest bound.