People have always had a religious or quasi-religious reverence for nature. In modern times, some people have started to see nature more as an enemy to be conquered than as a god to be worshiped. Such people point out that uncontrolled nature causes a tremendous amount of human suffering (to say nothing of all the misery that it causes other creatures), and that vast improvements to human welfare have largely been the result of us ceasing to love and fear nature and starting to control it.
There are several common responses to this. One response is that it is solipsistic for humans to measure the value of nature in terms of what is and is not good for us. This strikes me as right only insofar as it ignores the welfare of non-human creatures who have enough going on in terms of consciousness and/or sentience to matter; I think the objection would be without merit if one were to broaden the scope of concern to something like all creatures, present and future, capable of having experiences (who else is there to care about?). A second response is that seeing ourselves as highly effective lords over nature leads to dangerous overconfidence, which leads to costly mistakes in how we deal with nature. This is a very fair point, but what it really amounts to is a claim that we shouldn't underestimate the enemy, not that the enemy is really a friend. Anyway, the solution to that problem is to become better rationalists and get better at being skeptical regarding our powers, not to retreat into quasi-mystical Gaia worship. A third response is that getting into a "conquer nature" frame of mind puts people into a "conquer everything" frame of mind and leads to aggression against other people. This might have merit historically, but that problem is also best confronted directly, in this case by more effectively promulgating liberal humanistic values.
So what, if anything, is left to the idea that there is something special about nature worthy of particular regard? And by special I mean something beyond the fact that many people just plain enjoy it the way they enjoy lots of other things that nevertheless have no claim to any special status. I would say that the main thing that makes nature special in this sense is that when you are in nature or contemplating nature, you can be confident that the resulting thoughts and feelings are uncontaminated by all of the (visible and invisible) ideas and biases and assumptions that are present in your particular time and place. When you look at a waterfall and you like it, you can be pretty sure that: (i) it wasn't put there by anyone with an agenda; (ii) you weren't manipulated into liking it by contemporary ideology or social pressure or persuasive advertising or whatever; and (iii) the thoughts that you think while contemplating it aren't the thoughts anyone is trying to lead you into. In other words, nature is a way of guaranteeing that there is a little corner of experience that we are instinctively drawn to and that we can be confident doesn't represent anyone else's attempt to control us. And since other people are trying to control us all the time, even in relatively free societies (all the more so in oppressive ones), this is of real value.
I think the same basic point applies to some other things besides nature. Why do people still read old books* even when the knowledge in them has been refined and improved-upon in the meantime? In many subjects, we don't. Nobody learns geometry by reading Euclid, because there would be no point. But people do still read ancient works of philosophy. It seems to me that one good reason to do so is that for all the ways that these works have been analyzed and surpassed in the intervening years, the reader can be sure that what is written there is not the product of manipulation by the forces that are at work in the reader's own time and place. So it represents another way to gain valuable freedom and distance.
*Here I'm talking about non-fiction books. The merits of old creative works even when the innovations in them have become widespread in newer works is a different story. Often a point like the one in this post still applies, and sometimes the old stuff really is still just the best.
It's worthwhile quoting the summary given for the study you linked (this quote is not part of the study itself):
In other words, it's not being in nature that's good for us. It's just not being in a city that's good for us :-)
So once we learn to build super-duper-optimized resorts for relaxation, we can put them in every other city block, and your reason for preserving real nature will become irrelevant.
Apart from all the cultural views you quote, my animal core panics in nature. It thinks: There's trees and bears and rocks and, and different looking trees! Where's my car? My cellphone? My backpack of food? My insulin pump supplies? HEEEELP!
Don't get me wrong, I love a walk in a quiet wood. But to feel relaxed I need a lot of safety and infrastructure. I'd guess quite a few modern city dwellers are more like me than they are like "oh, nature, I know this".
The person who wrote this wouldn't last long in the wild.