VFD = Variable Frequency Drive, maybe Vacuum Fluorescent Display, occasionally if you mix those two up, it means Volunteer Fire Department.
AFD = Adjustable Frequency Drive, same thing.
VSD = Variable SPEED drive, could be a VFD, could be a DC drive, could even be a mechanical (Reeves) drive or an Eddy Current Clutch. Bottom line, the term VSD is not specific, VFD is. ASD is the same thing as VSD, Adjustable vs Variable.
A VFD uses line power as a raw material to make DC, it doesn't care if it's single phase, 3 phase, ABC rotation or CAB rotation. The output is 3 phase for the most part. But with single phase feeding a 3 phase motor, the incoming current increases by the sq. rt. of 3. Below 5HP at 230V, most VFDs on the market are sized for the additional power input capability to be used on single phase supplies. 5HP 230V and above, and 480V, most drives can be fed with single phase if the VFD size is 2X the motor size (but always size by FLA, not HP). A few drives cannot have their Phase Loss Protection disabled, making them unsuitable for this. Check your manual for a way to disable Phase Loss. If you don't see it, you can't use it for phase conversion.
There are a few versions of VFDs that work on only TWO versions of single phase motors, PSC (Permanent Split Capacitor) and Shaded Pole. Shaded pole motors can be speed controlled with voltage only, i.e. rheostat, which is a quantum leap cheaper. PSC motors are only good for lot torque applications like pumps and fans. Single Phase output VFDs are significantly more expensive than those tor 3 phase motors, so it only makes sense if you cannot replace the motor.