i've heard both so I checked.
From Wikipedia:
A bulletin board system (BBS), also called a computer bulletin board service (CBBS),[1] is a computer server running software that allows users to connect to the system using a terminal program.
So yes, you are correct.
The fun of BBS is the lack of privacy. You have to connect with your telephone number to someone else's phone number. If you thought your IP addresses was important to keep safe, imagine your phone number.
I'm half joking. I'm not sure but servers didn't have some caller ID program. I think it would show up on the phone bill though.