Need circuit for 12 volt 2 Amp step down power supply from 18 volt 1.5 Amp Input

Either can work correctly if designed properly. If you have a dumb rectifier supply feeding a 7805, then all the rectifier part needs to do is guarantee the minimum input voltage to the 7805 is met.

The problem is that such a power supply only charges up the input cap at the line cycle peaks, then the 7805 will drain it between the peaks. This means the cap needs to be big enough to still supply the minimum 7805 input voltage at the worst case current drain for the maximum time between the peaks.

The advantage of a full wave rectifier is that both the positive and negative peaks are used. This means the cap is charged up twice as often. Since the maximum time since the last peak is less, the cap can be less to support the same maximum current draw. The downside of a full wave rectifier is that it takes 4 diodes instead of 1, and one more diode drop of voltage is lost. Diodes are cheap and small, so most of the time a full wave rectifier makes more sense. Another way to make a full wave rectifier is with a center tapped transformer secondary. The center is connected to ground and there is one diode from each end to the raw positive supply. This full wave rectifies with only one diode drop in the path, but requires a heavier and more expensive transformer.

A advantage of a half wave rectifier is that one side of the AC input can be directly connected to the same ground as the DC output. That doesn't matter when the AC input is a transformer secondary, but it can be a issue if the AC is already ground-referenced.

I understand why you see the initial fast droop, because the electrodes in the battery are quite complex electrically:

^{simulate this circuit – Schematic created using CircuitLab}

At the application of the load, an EDLC forms at the electrodes and the capacitance can be enormous - 60μF to 100μF is not unusual (the effective plates are literally Angstroms apart) with a sub milliohm ESR and that gives you a very effective differentiator at initial transient loading of the battery into a heavy load.

Once the EDLC has formed, it stops conducting and the only resistance is the ESR of the battery electrodes themselves. As the terminals recover to being limited only by load and internal ESR, there can be an inductive effect (after all, self inductance exists anywhere current flows, to a greater or lesser degree).

Your waveform looks like a damped resonance, so an appropriate snubber may work.

You can see how EDLCs form at battery electrodes in this paper.

I have seen this effect a few times and it was quite confusing the first time it appeared (in a seawater activated circuit).