POSIX first was a standard in 1988 long before the Single UNIX Specification. It was one of the attempts at unifying all the various UNIX forks and UNIX-like systems. POSIX is an IEEE Standard, but as the IEEE does not own the UNIX® trademark, the standard is not UNIX® though it is based on the existing UNIX API at that time. The first standard POSIX.1 is formally know as IEEE std 1003.1-1988. IEEE charged a substantial fee to obtain a copy of the standard.
The Open Group released the Single UNIX Specification (SUSv2) in 1997 based on IEEE's work of the POSIX standard. SUSv3 was released in 2001 from a joint working group between IEEE and The Open Group known as the Austin Group. SUSv3 is also known as POSIX:2001. There is now also POSIX:2004 and POSIX:2008 which is the core of SUSv4. As for what UNIX® is, UNIX® is whatever the current registered trademark holder says it is. Since 1994, that is The Open Group.
Novell acquired the UNIX® systems business from AT&T/USL which is where UNIX® was born. In 1994, they sold the right to the UNIX® trademark to X/Open now know as The Open Group. They then sold the UNIX® source code to SCO as UNIXWARE®. UNIX® itself has forked many times partly due to AT&T's licensing model. Purchasing UNIX® gave you the complete source of the operating system and the full tool-chain to build it. Modifications to the source can be distributed and used by anyone who owned a license to UNIX® from AT&T. The license fee was in the thousands.
BSD was a project at Berkeley which added a number of enhancements to the UNIX® operating system. BSD code was released under a much more liberal license than AT&T's source and did not require a license fee or even a requirement to be distributed with source unlike the GPL that the GNU Project and Linux use. This has cause a good part of the BSD code to be included with various commercial UNIX forks. By around 4.3BSD, they had nearly replaced any need for the original AT&T UNIX® source code. FreeBSD/NetBSD/OpenBSD are all forks of 4.3BSD that are a complete operating system and have none of the original AT&T source code. Nor do they have right to the UNIX® trademark, but much of their code is used by commercial UNIX operating systems. The Socket API used on UNIX was developed on BSD and the Unix Fast Filesystem code was borrowed and used on various UNIX Operating Systems like Solaris with their own enhancements.
Linux was developed in 1991, but was developed from scratch unlike BSD and used the existing GNU Project which was a clean-room implementation of much of the UNIX user-space. It implement much of POSIX for compatibility and is UNIX-like in design, but it does not have the close connection to AT&T or UNIX® that the BSDs have.