An appliance with the primary role of storage, and the ability to share files over the network are the distinguishing features for a NAS. Network support for iSCSI, SMB, NFS makes a NAS; sharing data exclusively over the media protocols (http, rtsp, etc) makes it a media server
By "NAS" most people mean "box they bought from someone to share files on a network" - think Qnap or Synology. They'll call a "NAS" that is home-built a server, even if they do basically the same thing.
To make it more fun, you'll have people refer to "I don't have a NAS, I run FreeNAS on my server."
> My home server has a ZFS array, the media server and some other programs access it directly.
This sure sounds like a NAS.