Thunder in Paradise has it pretty much correct. Cart-based console games are dumped using copiers (Neo-Geo cart used a backup device called Multi Game Doctor 2). Arcade boards are dumped with devices similar to EEPROM readers, but they aren't EEPROM readers per se. (At least, not in all cases). Backup devices can also send rom data directly to the console, making it possible to play the dumps directly with your console.