Unfortunately, I think your definition is rather self-referential, using the term to define itself, and it still leaves some important questions unanswered. You can say that it's the nature of rivers to run. It's also the nature of rivers to back up when they are dammed, and when water is released from that dam, it naturally produces energy as it seeks a lower level. So is a dammed river part of "nature" or not? For that matter, nuclear fission also occurs in nature.
My point, if you haven't guessed it, is that "nature" is a friendly-sounding, comforting term...but as we can see with the items on our supermarket shelves, labeling something "natural" doesn't always make it good. Not only can the meaning be distorted, but there are plenty of natural phenomena (syphilis, earthquakes, tsunamis) that do harm.
Get where your coming from. Yes it does sound and can be easily taken that way.
I would expand on my definition this way: Instead of natural laws I could say the underlying natural laws. Therefore in your example of rivers following the laws of nature then it would mean they are bound by and follow the laws of physics.
Once I got into a conversation with someone who insisted something was normal. I said it may be normal but is it natural?
In that particular discussion I was looking at what underlying principles or 'unmanmade' laws it fitted with or went against, this thing called normal.
Just interesting mental exercises, I find anyway.