If my research is applying methods from field X to field Y, should I do my PhD in field X or field Y? Let's say I'm a PhD student in a field, let's call it X, and my research is all based on taking the methods developed in field X and applying them to some other fields, let's call them Y, Z, and W. Let's further suppose that I do not make any contributions to the theory of X, I only apply it to Y, Z, and W. Is X the proper fit for me, or should I really pick my favorite of Y, Z, and W?
A PhD is meant to extend the knowledge within a field. It's obvious that you can contribute knowledge to Y, Z, and W, at least if you apply the X methods to solve some research questions in these fields. It's less clear what you contribute to X.

You state that you don't contribute to the theory of X, but maybe you extend or generalize the X methods alongside applying them? If you do that, you could get a PhD in X. Otherwise, Y, Z, or W would seem to make more sense.