The fact that dc was originally intended as basically the Unix system's bignums library http://www.catb.org/esr/writings/taoup/html/ch08s02.html#id2... helped to open my eyes to this. I think you could talk about this idea of Unix as "big Unix" and the narrower understanding of Unix programming, dominated by the C APIs, as "little Unix".
Little Unix took over the world, while big Unix collapsed under its own weight due to things like portability/versioning hell, robustness problems due to faulty tools and text-parsing errors, performance issues and so on. (Partly this was a success catastrophe, of course, for example in how the popularity of Unix resulted in a profusion of different Unix userlands with varying interfaces and bugs.) Also little Unix was more comprehensible from, and more portable to and from, the world of PC (MS-DOS/Windows/Mac) application development, where the environment is more or less the kernel's/OS vendor's APIs. The fact that clearly many more people have bought or read K&R http://www.amazon.com/dp/0131103628/ (with its very little-Unix perspective) than /The Unix Programming Environment/ http://www.amazon.com/dp/013937681X reflects this divergence, and also surely helped to create it.