Then you must have skipped over the whole part about how the "ROM" in P/E/EEPROM has nothing to do with the read-only aspect of ROMs, only the ability to hold data without a current. So get over his "read-only" fetish you have. It doesn't apply.
The devices are technologically different and have no relation to each other except that they 1) hold data without current, and 2) reside on a chip. All of the uses and locations you describe with varying degrees of ease of writing have nothing to do with the issue, which is that a PROM is not just a programmable ROM, nor is an EPROM just an erasable PROM. A ROM cannot be made programmable, and a PROM cannot be made erasable -- they had to come up with completely new products with completely new technologies to accomplish that.
ROM != PROM != EPROM != EEPROM. EE/E/PROM is not a subset of the set ROM, EE/EPROM is not a subset of the set PROM, EEPROM is not a subset of the set EPROM, and none is derived from another. They are all distinctly different sets, although all are a subset of the set "chip that holds data without current."
You still haven't told me why we don't call an OTP EPROM a PROM, since the use is exactly the same.
Why do you keep repeating your errors?
ROM is an acronym for "Read-Only Memory"
Are PROM's Read-Only Memory? Yes.
Are EPROM's Read-Only Memory? Yes.
HINT: The ROM in PROM and EPROM means that same thing as ROM.
What you are attempting (and failing) to argue is the way ROM's and PROM's are implemented is different therefore the implementation of PROM's are not equal to the implementation of the ROM's but you are wrong since the acronym is in no way trying to address how the memory in implemented - only how it is used.
Spend a bit more time thinking and less time doing victory dances.