Definition of antenna efficiency
The first is a more general definition, the basic statement of what efficiency means.

The second can be derived, if you model the antenna as a radiation resistance Rrad, in series with an Ohmic loss resistance, Rohm.

Reactive power is not considered in these equations. Why should it be, over a full wave it contributes no net power flow.

So I think the mistake in your argument is asking why P doesn't include the reactive power in the antenna. It doesn't need to; only the power transferred to the antenna should be considered.

(and for any alternating current, P means of course |P| integrated over a cycle, not instantaneous power. Otherwise you'll quickly get tangled in reactive power questions)