Though I understand, and in a way agree with your point—that human changes to the environment are as natural as that of beavers—, I think one important difference is the scale of those changes.
Ours are not as natural in the sense that we no longer rely on our inborn capacities to produce those changes.
Yes, in a way we are like beavers in our engineering. But we are insanely overpowered beavers. The scope of our changes puts us in a class of our own.
So if we call changing the environment by one's own means only 'natural', and massively changing it by compounded means 'artificial', we are not natural, not as beavers. Ours is artificial.
So contrary to what you said, that distinction makes at least some sense.
Modern human civilization is the apex predator that still often feels and acts like it is prey on the African veldt. So you get insanely overpowered beavering while some of us still compare us on the same level as beavers.
Yes, I am definitely suggesting wolves (not "wolfs") did not build ships or airplanes and migrate around the world in an astonishingly short amount of time. They migrated slowly, on foot, as they were able to by following food sources, and using land bridges since wolves can't cross oceans.
Same goes for viruses (I'll give you a pass on that one, it's more accepted now than "virii"): they don't float across oceans in a week, unless they're carried by birds (which most aren't).
Beavers build dams, flooding vast areas and it’s natural.
When humans do it, it’s not.
Makes zero sense.