I don't recognise the subscripts used in your equations, so am having trouble identifying what they stand for.

Electric and magnetic polarization currents as in the below article.

I. Liberal and R. W. Ziolkowski, “Analytical and equivalent circuit models to elucidate power balance in scattering problems,” IEEE Trans. Antennas Propag., vol. 61, no. 5, pp. 2714–2726, 2013.

The first equation looks for the extracted power by integrating all the places where the incident field is in phase with the current density. There should be some current on the dipole in phase with the incident field, so this shouldn't give a zero result, even though most of the current on the dipole is not in phase with the incident field, because it represents stored energy. Have you properly included the current on the dipole in the current densities? If it is represented as a surface current the density may be infinite and the volume zero, so the result may depend on the way you worked that out. If done correctly I would expect this to give the sum of the received power and the scattered power.

The second equation perhaps integrates the power in the termination. Is Et the field in the termination?

I think the third equation integrates the products of the scattered fields with the current densities. Does this give a result? Again it relies on including the surface currents correctly.