Half wave vs full wave rectifier
The 4 diode bridge is a full-wave voltage doubler and achieves 200% of the voltage and 50% of the current of the centre-tapped 2 diode full wave version. Thus the tapped version has lower impedance.
Some users might consider the tapped 2 diode version as a 2 phase each half wave combined to make a full wave. But in fact, the secondary is only a split-single phase.
A half-wave bridge is a single diode version is used when less voltage and current is needed and thus the longer charge interval is adequate.
With a full-wave bridge rectifier you have four diodes, resulting in two diode drops, and the capacitor value is roughly half what you need for a half-wave rectifier with the same ripple. If you have a center-tapped transformer you can use two diodes to get full-wave rectification with only a single diode drop, but the extra transformer tap and associated connections are not free. For rectifying mains voltage or voltage from a typical AC-output wall plug adapter you don't have the option of a center tap (usually).
The four-diode bridge uses a transformer capacity most efficiently, center-tapped (two diodes) full-wave is better than half-wave but worse than the four diode bridge for voltages much higher than a diode drop.
With a half-wave rectifier you have only one diode, and only one diode drop, but you need a larger capacitance (typically physically about twice the volume). It's the least efficient way to use a transformer (as a single rail anyway).
Most switching power supplies use full-wave rectification on the input. On the output they may use half or full wave as the higher frequency means the capacitor size is not as important. With a flyback converter you don't have a choice- it has to be half-wave.
Half wave rectification is sometimes used with an AC-output wall plug adapter to produce balanced +/- rails for audio. I have a hydrogen-oxygen electrolysis welder (US made ca. 1968) that uses half-wave rectification probably because the optimum voltage for the gas production cell is only a couple volts and two fat stud-mount silicon rectifier drops would be less efficient and more costly even though the current is many amperes (no filtering required in that application). These days we'd probably use a switching supply with Schottky diodes at the output.
The half wave rectifier uses one diode, does not need center tapped transformer. The full wave one uses two diodes and needs center tapped transformer. Efficiency for both cases are the same. At the same load with the same output filter design, full wave rectifier has less ripple. From another perspective, with the same ripple requirement, filter design is easier for full wave rectifier with less output capacitance needed.
For the very light load design, half wave rectifier is more suitable, simple and cheaper.