There is no single way to evaluate a source for reliability or appropriateness. Instead, we must ask many questions of each source. Below are some ways we can look at a source to evaluate it.
"Information resources reflect their creators’ expertise and credibility, and are evaluated based on the information need and the context in which the information will be used." 1 Framework
Whether something is important or can be viewed as an authority depends on the situation you plan to use it in. A tweet about a political policy isn't generally something we would cite. But if the president is the one tweeting it, how does that change the context? A famous speaker might lend authority to a particular management technique, but if they are trying to sell you training materials on that technique from a company they own, does that change our trust in their authority on the topic? If a peer-reviewed article uses an experiment to support its hypothesis, but the experiment was funded by a corporation with a conflicting interest, does that change how reliable we think it might be?
Where the actual newspapers are placed here is of less importance than the two axes. Note that complexity and bias are separate! As for the actual placement, this is just one way of doing it. It's your job to evaluate specifics for yourself.
When looking for research on a topic, remember currency affects what format it may be published in. If it has just happened there will be no peer-reviewed, analytical sources. Can you instead use a current news source and an older peer-reviewed source on a similar topic?
Every time we evaluate a source we must ask these questions of it--news source, journal article, website, or something else entirely--to determine whether it is a source we can trust to serve the necessary purpose.