It doesn't matter. The chip is writable, period. Only one type of chip is read-only, and that's a ROM. If you were right I could tell you my EEPROM USB key is a ROM because I flipped the write-protect switch. Usage and accessibility doesn't change what it is.
No it is not. An EPROM in a computer IS NOT WRITABLE - Period - End of Story.
In a computer, an EPROM is Read-Only Memory - one can not write to an EPROM in a computer.