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.

I find any use of the same word to mean opposite things confusing, and it seems to happen more often than I would like in the US politics

word usages change

Maybe it changed in your country but that doesn't affect the meaning it has in my country. We may use the English language to communicate, but we're not reproducing all of american culture, and we don't bring over all of their changes. Yes, it's confusing for you. For us too. No, don't expect the rest of the world to adopt all the changes you make.