There is a misunderstanding of the terms here.
EEPROM refers to electrically erasable programmable read only memory and usually has not that much memory. Also for as far as I know only used for some parameter storage in vintage synthesizers, and at that not even in the really vintage ones.
EPROM stands for erasable programmable read only memory and refers to, amongst others, the old 27xxx range of IC's that have these nice glass windows to allow erasing with UV light. These are commonly used in the old synths.
Parameter or patch storage was mostly done with battery backup of the RAM making it non volatile.
Later on FLASH devices came along and where used to store firmware.
So it depends very much on the synth you are looking into.
And like Ian.m wrote the AT27C512R-45PU is OTP, which means you can only program it once.
Be aware that The other programmer you linked to, is
not capable of programming 27Cxxx series that need 25V for programming the device.
Support high-voltage chips, such as 27Cxxx series, VPP Maximum up to 25V, that is what TL866II cannot achieve.
Edit: I read it wrong, it seems to be capable of doing it because it refers to the TL866II not being able to achieve the 25V.
Do make sure that the devices you need to program work with the 25V this programmer can provide. Older types might need 28V.
Edit2: Here is some background info to be found.
https://cpushack.com/EPROM.html