You're both right. FoxNews channel is the #1 cable news station, but I am pretty sure they still carry the highest viewership ratings of all news programs. The main distinguishing feature between them--as a cable news network--and the alphabets is that viewership for any given news program, and TV news overall, is way down compared to the halcyon days when you and your parents cozied up to an evening with Murrow or Cronkite like a dependable 80% or so of the TV viewers would.
Fox's claims are dubious in the sense that they use their viewership as some metric for a changed political demographic or even strangely, "claiming that the majority of people think like we do because we are the most popular." That is John Conner level thinking. Back when there were only 3 stations to choose from (maybe 4?), ratings needed to own a good 40%+ share for any given program to be considered successful, and *I think* that was on the low end. I love Lucy owned something like an 80%+ share for their time slot, iirc. Murrow/Cronkite were no different.
Today, Fox wins out with something like 20% of the audience, but of course that is because of massive dilution with choice. It certainly is no metric for claiming that they have a majority share of the public opinion, that they are the most trusted--but they are the most watched. These days, when looking at 24 hour news viewership, it simply seems to mean that they are "the most entertaining," a distinction that they are very proud to carry.