the modern American use of "liberal" as a synonym for the progressive left... but word usages change

Modern American civics education is, on average, basically non-existent. All changes in word usage trends toward less nuance, greater redundancy, and greater ambiguity.

And that's fine, I guess, because Descriptivism or whatever. But in the event that moderately educated people use the terms "Liberal" or "Leftist", it's probably okay to assume the formal meanings.

makes references to the "liberal media" confusing."

You seem to not be that confused, since you correctly pointed out that these companies are owned by conservatives.