Many or all comets visible to the naked eye are referred to as Great Comets of their year(s). Might NEOWISE too be called like this? Why or why not? Is the IAU responsible for declaration of Great Comets or another organization or no such body?
I'm inclined to say a hard no using the casual definition of a "great comet"
A great comet is a comet that becomes exceptionally bright. There is no official definition; often the term is attached to comets such as Halley's Comet, which during certain appearances are bright enough to be noticed by casual observers who are not looking for them, and become well known outside the astronomical community.
So, there's a dual definition of visible brightness and/or recognition outside the astronomical community. With that in mind, I think it's pretty clear that Comet NEOWISE doesn't make the cut due to it not being easily visible Granted, it peaks in 2 days from now (7-23-20) but so far it's hard to see by the naked eye and that's not likely to change significantly as it peaks.
Although it’s brighter than all but around 20 stars, its extended, diffuse nature makes it a challenge for human eyes.
NEOWISE is the brightest comet since comet McNaught, but McNaught, with a peak mangitude of -5.5, was about 500 times brighter. (based on the estimate that Neowise is about as bright as the 20th brightest star, roughly 1.25 magnitude). McNaught was actually brighter than peak Venus, though it probably didn't look that bright to the naked eye because it was more spread out.
Comet Lovejoy was a bit unusual. It's "greatness" had nothing to do with it's visibility, which was unspectacular. Lovejoy passed too close to the Sun and it lost too much material in passing, so it was much anticipated and I remember articles saying it was potentially a super-comet, but it lost everything on the other side of the sun and we never got a good view from Earth. Perhaps if we'd been on Mercury or Earth's L3 trojan point, we'd have seen a rare and magnificent show.
Wikipedia doesn't list Comet Lovejoy in it's list of great comets.