A DC generator produces DC directly
An Alternator produces AC which is then rectified to DC with diodes, Most common alternators are three phase AC at a variable frequency which is directly related to the r.p.m. of the Alternator.
An AC generator produces AC directly from the coils, the frequency of the AC is directly related to the shaft speed and the number of coils on the generator.... Common AC generators are 3600 r.p.m. or 1800 r.p.m. depending on the number of coils , 2 or 4 are most common
Honda Inverter generators are a hybird.....they are 3 phase alternator (AC) which is rectified to DC at about 80 volts which is then used to power the inverter to make the AC. So they are AC converted to DC then back to AC.
The advantage is frequency output is not directly coupled to the R.P.M. of the engine.....thus the inverter generators change speed to be more effecient which a AC generator cannot do as frequency is directly related to r.p.m.
the frequency of the alternator is directly related to the speed of the engine, the frequency of the inverter is derived from the logic in the alternator. They do not need to be the same.