On the "streets" there are no internal drives with enclosure. Once is in a enclosure, it automatically is external, but am sure somebody is gonna invent a new term.
For your purpose a backup SHOULD be external, you wanna keep your backup at some distance from the box that you are backing up from. If you wanna go "purist" about it, folks even keep their backup drive OFF SITE in case of theft/fire etc. If you have USB3, that's clearly the interface of choice, but USB2 is OK too if you are just gonna turn on the backup and let it run overnight. To keep the drive cool, am fan (no punt intended) of those enclosure that looks like a big heatsink, with fins, they are called passive cooling and rid the necessity for a noisy fan.