I see links identified as a "partner site" all over the place, I read posts of site owners looking to "partner" with another site, etc.
Is there a specific meaning to the term "partner" when used in this context, or is it a vague term that means whatever the user wants it to mean? I suspect the latter.
it means that they exchange traffic, thats about it.
Partners has different meaning.
It can be one or more of the followings:
Content exchange partner
Link exchange partner
Technology partner, one offer a technology and another give them a power by statment or other benefits
email subscription/promotion partner
Revenue sharing partner
As Addto pointed out, the term "partner site" can have many, many meanings. You also might hear "affiliate site" and "alliance site", etc. There is no correct or incorrect way to use these terms, it all depends on your perspective.