I like what you shared. I actually can't believe I haven't seen that paper before (it has >4k citations ffs), but I admit, I work in a less statistical side of physics.
What I'll say is that what something is is a fuzzy term, although I generally agree what you mean especially about a power law says about the behavior in the tails especially. That comes from thinking a small deviation on the tail is "small" due to optics and not realizing it is deviation on orders of magnitude. That might be what the replier to my comment was referring to. That's why "two graphs" is usually a good answer.
PS
Also, I would never fit a straight line on a loglog or semilog graph, nor should anyone ever. People who do that don't understand what least squares is at all.
What I'll say is that what something is is a fuzzy term, although I generally agree what you mean especially about a power law says about the behavior in the tails especially. That comes from thinking a small deviation on the tail is "small" due to optics and not realizing it is deviation on orders of magnitude. That might be what the replier to my comment was referring to. That's why "two graphs" is usually a good answer.
PS Also, I would never fit a straight line on a loglog or semilog graph, nor should anyone ever. People who do that don't understand what least squares is at all.