The reason I did this was because it is conventional to write the low temperature result of Debye theory in terms of the Debye frequency. But I derive the low temperature result without talking about the cutoff at all, and only later do I introduce the cutoff. So instead, I just define some particular bunch of constants to be the Debye frequency. Then later when we need to implement the cutoff, we show that this particular set of constants is in fact the right cutoff frequency to choose in order obtain exactly 3N modes. Does that make sense?