Who is right depends on where the drive is attached. If it's attached to the Mac, I think it can be included in the backup. If it's attached to a router, it cannot.
Correct. A drive attached to a router becomes a NAS.
The term "External Drive" refers to a drive directly connected to your computer. Another term is DAS.
By contrast... NAS drives are drives connected to the network (either via a router, or built in NAS functions, or through another computer. They cannot be backed up to Time Machine as far as I know.
/Jim