Guide to Unix/Variants
There are many variants of Unix, both commercial and free/open-source (FOSS). Some of the most widely deployed variants are Linux, on the server side, and macOS (formerly Mac OS X), on the desktop side. Android has a Linux kernel and is a widely used mobile operating system, yet far from being a full Unix variant for its lack of command-line utilities and other features. MINIX is a variant that is running internally in Intel chipsets post-2015.
Standardization efforts resulted in creation of POSIX and Single Unix Specification (SUS) standards to ensure minimum interoperability between variants, but only a handful of Unix variants are currently registered as fully complying, and indeed, many variants including Linux variants are not fully complying with SUS.
As for trademarks, UNIX (so capitalized) is a trademark of The Open Group, a consortium of companies that publishes the SUS that forms a basis for certification. Only variants licensed by The Open Group to use the trademark are allowed to use it. Historically, the lack of availability of licensing the trademark is what contributed to the proliferation of different names of Unix variants. POSIX is a trademark of the IEEE.