Differences in full wave rectifiers
Personally I wouldn't consider your final circuit to be a full wave rectifier at all.  It's two half wave rectifiers the first charges C1 during the positive half cycle and the second charges C2 during the negative half cycle.

This circuit does indeed double the voltage and a slight variant of it used to be common in consumer electronics. As shown below.







simulate this circuit &ndash; Schematic created using CircuitLab

With SW1 open the output is close to the peak of the AC input voltage and all 4 diodes provide full-wave rectification. R1 and R2 are needed to ensure that C1 and C2 share the voltage equally.

If Sw1 is closed then D3 and D4 do not conduct and R1 and R2 are not doing necessary but the output voltage roughly twice the input AC.

Bearing in mind that European mains voltages are typically 220V or 230Vrms while in the US its 120Vrms.  This circuit provided an easy way to make devices work anywhere in the world just by configuring SW1 to be open or closed as appropriate for the country it was going to be used in.

It's less common now as for low power devices it's relatively easy to design a wide range input fly-back converter and for higher power units devices typically need some form of power factor circuitry.  I'm sure it still does get used occasionally though.