I've never heard of the weekly world news. The National Enquirer from what I seen at the check out counters is for UFO conspirators and Elvis sighters. Now I don't like the term fake news at all. It's a illegitimate portrayal of biased and slanted news coverage. Since I don't subscribe or read either the NYT or WSJ, personally I can't tell you whether one, the other or both fall into the biased or slanted news coverage. I have however heard the NYT's does and is on the side of the democrats/liberals in their reporting. Heard is second hand, unproveable by me.
I can tell you my own AJC is indeed biased and slanted in their news reporting heavily toward the democrats/liberals. Atlanta is a very liberal city where perhaps half the liberals in Georgia reside in and around the Atlanta metro area. So to be slanted or biased in a liberal way makes sense since the AJC is in the business of selling newspapers. Give the customer what he wants in a way he wants it. fake news, no. Biased news, yes.
The same holds true for the cable news networks. Give the views the news in a way they want to see and hear it. Fox from a conservative point of view, CNN and MSNBC from a liberal point of view. Again their job, business is to attract and keep viewership. The more viewers, the more each can charge for their commercials, the more money each can make. Making money is what these three corporate run business new networks are all about. Again, slanted and biased news reporting, but from a more business oriented position or reason. Making a profit.
There are times I wonder about this big hoopla about news reporting. Most Americans don't trust the media to report the news accurately, fully or fairly today anyway. According to Gallup, that trust has fallen from 72% back in 1976 down to 32% at the end of 2016.
https://news.gallup.com/poll/195542/americans-trust-mass-media-sinks-new-low.aspx
I think most Americans recognize the fact our news media has taken sides in our political battles and as such, their trust in receiving accurate, fully reported news in a fair way has dropped. This is the news media's own doings. But they probably make a ton of money being biased and slanted as that as a whole is what each of us want. especially the more partisan of us. News that reinforce our partisan and ideological views.