Of two conflicting desires, we call second-order the one we don't expect to go away, and as such more invariant, part of self, even if not ever in control.

Most second-order desires are not about first-order desires, they are about the same thing as the first-order desire. For the second-order desire, modifying the first-order desire is instrumental, not terminal, and the same applies in the other direction. The differences I see come from first-order desire being actually in control, and being stupid enough not to work on eliminating the second-order desire.