Installing custom ROMs: since most Android systems don't have a ROM, isn't this a misnomer? My Samsung Galaxy SIII seems to have an internal SD card as internal memory,
and no ROM (Read-Only Memory).
As far as I know, ROM refers to a type of memory which can be only modified by attaching a cable from it to a special device and flashing; once the ROM is flashed, it is for all purposes read-only from the point of view of the users logging into the OS. Many sites report that one of the benefits of rooting is installing custom ROMs.
So why is installing an alternative system UI such as CyanogenMod still referred to as installing a custom ROM? Isn't it just replacing some files in the /system directory with some other files, i.e., deleting or backing up some old files and copying the replacement files to that directory?

Thanks.
Old things tend to stick around. In everyday things (toasters, microwaves etc) there is some kind of software to control the device. That software is installed by the manufacturer and is usually hard to change, in practice read-only. While you can flash the system partition on android, with some electronic devices you needed to and still need to plug in some specific pins to some manufacturer application on a pc.

Nowadays ROM refers to the software by the manufacturer that drives the device, such as the /boot and /system partitions.