Local magnitude, the Richter scale
USGS says the logarithmic magnitude idea was developed by Charles Richter in the 1930s for earthquakes in southern California, using relatively high-frequency records from nearby stations. That scale is ML. The L stands for local. It is the scale that came to be called the Richter magnitude. The same FAQ says the method is strictly valid only for certain frequency and distance ranges.
Body-wave and surface-wave magnitudes
As seismograph stations spread, USGS says new scales extended Richter’s idea: body-wave magnitude (Mb) and surface-wave magnitude (Ms). Each is valid for a particular frequency range and a particular kind of seismic signal. Inside its valid range, each is meant to be equivalent to the Richter magnitude. Outside that range, it is the wrong tool.
Moment magnitude
USGS says ML, Mb, and Ms are limited, and that for very large earthquakes moment magnitude, Mw, gives the most reliable estimate of size. Moment is a physical quantity proportional to the slip on the fault multiplied by the area of the fault that slips. It can be estimated from seismograms and also from geodetic measurements, then converted by a standard formula into a number similar to the other magnitudes. USGS says Mw is valid over the complete range of magnitudes, which the earlier scales were not.
Where the factor of 32 lives
The separate USGS page on magnitude, energy, and intensity is the one that states the tenfold amplitude step and the roughly 32-times energy step. That arithmetic is the magnitude and energy tool and the magnitude versus intensity page. This page is only about which scale the number came from. A scale name is not a forecast.