PROM is just OTP?

Well, literally, "PROM" means "Programmable Read-Only Memory" and "OTP" means "One-Time Programmable"
Thus, if a PROM can't be reprogrammed, it must be only One-Time Programmable.

So, taking the words at their face value, "OTP" and "PROM" are equivalent.
However, "OTP" is usually taken to indicate the specific case of a device which would be UV-erasable if it weren't for the fact that there was no way to actually get the UV light onto the chip.

"or it includes EPROM and EEPROM?"
They're all types of PROM; ie, Read-Only Memories which can be Programmed.
It depends on your application whether being Erasable or not is significant and, if so, whether UV or electrical erase matters; eg,
If you're just thinking of a processor which needs somewhere to read its code from, then the processor doesn't care if it's Programmable or Erasable - it just needs to read code from it.

However, if your system needs to support code download, then you must use some form of electrically-erasable PROM - unless you want to build a UV lamp into your product! ;-)
(Normally, one would choose Flash here)

"Also, EPROM can be erased by UV light, then how to erase EEPROM?"
Remember, "EEPROM" stands for "Electrically-Erasable PROM"
Thus the erasure is done electrically, rather than by UV light.
The device datasheet will specify precisely how this electrical erasure must be done.

"Can I erase an EEPROM by a universal programmer such as SuperProIII?"
The way to answer such questions is always: Read the manual!
Note that different EEPROM chips have differing erasure requirements; the manual for your particular programmer will tell you whether it can handle a particular EEPROM device.