When I said it was not zero, I meant it was not zero in terms of the way the circuit is connecting when analyzing a circuit loop and determining if it is a series- or parallel-resonant circuit. Your claim that the top of the inductor is connected to the bottom of the capacitor through this so-called zero impedance is laughable.
Batteries and other voltage sources do have an internal impedance (which is ideally zero, but never really is; and sometimes it's not even low), and this fact is important when attempting to transfer maximum power from a voltage source to a load, but it is not considered as a part of such an analysis as this.