The naming is interesting
But, personally I use UNIX for UNIX, and Unix for UNIX-like. Linux is a kernel not an operating system, once you realise that, then GNU/Linux makes more sense but easier to say Linux distro or Linux System. If you just say Linux and don't supply context then it is assumed you are talking about the Linux kernel.
UNIX is very POSIX, and the BSDs are all quite UNIX-like, but Linux is part of the Unix family.
See, that looks quite neat.
There are a few books that use the Unix name, such as Unix Shell Programming and it would look a little daft to have UNIX-like scattered around there. Absolute BSD brings forth the UNIX-like idea, and the older books tend to use UNIX for UNIX.
The history is actually quite illuminating, coming from MULTICS, UNIX was a one user stripped down version of it. So whilst it came from multi user, it actually was a single user operating system running on a PDP-7, and hence was actually called UNICS to mean a castrated MULTICS a pun on Eunuchs, the spelling was later changed to UNIX.