Her normal speed pronunciation sounds like Le. But when you hear the slow version it clearly sounded like La.
Agreed. I got it wrong twice because the normal speed audio does not sound correct. (Now my ability to remember this detail is, admittedly, my own problem.)
Please remind me if this can ever be translated, "the media" (such as the collective group of news people or the form of mass communication), or if that is a different french phrase/word?
"La presse" is the press in the sense of "the written media", "the papers" (newspapers, magazines, and so on). An individual magazine is "le magazine".
I listened to it half a dozen times. It sounds like she's saying something like "le preuf" — very far from an S-sound.