From the section you quoted:
"It (natural law) does not refer to the laws of nature, the laws that science aims to describe. "
When moralists use the term "natural law" they are not (or should not) conflate what they are saying with science.

All "laws of science" are inaccurate and doomed to be at least potentially subsumed by some grander law.

From your 1st physicsworld link:
"WHAT do we mean by "the laws of nature"? The phrase evokes a set of divine and unchanging rules that transcend the "here and now" to apply everywhere and at all times in the universe. The reality is not so grand. When we refer to the laws of nature, what we are really talking about is a particular set of ideas that are striking in their simplicity, that appear to be universal and have been verified by experiment. It is thus human beings who declare that a scientific theory is a law of nature and human beings are quite often wrong."

Nevertheless, if a law is actually broken, it is no longer a law.