Thanks for elaborating, I think I understand what you mean now! I was really confused when you said that beaver dams are just like fences, because to me they're not at all the same. It sounds like we actually have a very friendly disagreement. =)
I think "nature" is our best way to express these important but nebulous concepts, but you think it hides those exact same concepts. We're probably both right -- it will depend what community we're in, and their understanding and interpretation of the word "nature".
That aside, I agree with everything you've just said. The only point where we differ, I think, is that of whether the word "nature" and its allies make a useful shorthand for any of these concepts.
I think the points you've raised deserve more focused consideration than that. For example, I agree that it's far too easy and thus too common for people to forget that we do have a unique power over our environment, and to use that power in ways that do harm. I don't think that deserves to sneak past under so fuzzy and easily dismissible a term as "unnatural"; I think it deserves to be called out for exactly what it is.
That's a lot harder to do when we make a habit of glossing over such considerations with a shorthand like "nature". That's undesirable because these are the considerations that inform every decision we make about how we exist and behave in our environment, on scales as large as that of climate change and as small as what we do with our backyards, when we have those. Or, for that matter, whether we have those. None of this stuff is as simple or as abstract as "nature" makes it seem, and that's why I argue that "nature" fails to serve a useful purpose.