You're right, and both definitions tend to be used interchangeably. I'll work on correcting that in my own speech, but I think in the meantime here's the essence of it:

Privilege is a phenomenon that occurs as the result of a special kind of status, but the term also gets used to describe the form of status that generates the phenomenon.

When a status based on group identity is pervasive enough to be invisible to members of that group, the resulting assumptions lead to a set of behavior called "privilege". It's probably even clearer to use the term "privileged status" than mere "privilege", when talking about the status itself rather than the resulting behaviors; I'm going to try using that myself for the next few weeks and see if I can anchor some critical self-analysis to the process.