From The Art of Unix Programming (emphasis added):
After the [1974] paper, research labs and universities all over the world clamored for the chance to try out Unix themselves. Under a 1958 consent decree in settlement of an antitrust case, AT&T (the parent organization of Bell Labs) had been forbidden from entering the computer business. Unix could not, therefore, be turned into a product; indeed, under the terms of the consent decree, Bell Labs was required to license its nontelephone technology to anyone who asked. Ken Thompson quietly began answering requests by shipping out tapes and disk packs — each, according to legend, with a note signed “love, ken”.
There is much more relevant information in that chapter; its title is "Origins and History of Unix, 1969-1995". Highly recommended reading (along with the rest of the book!) :)
Unix was making that distinction since the very beginning, i.e. since version 1 was released in 1971.
The system was booting to multi-user mode (i.e. users connected to the available serial interfaces, tty0 to tty5 but provision was made to add four more ttys).
Unix v1 manual states for the section 4, tty page:
By appropriate console switch settings, it is possible to
cause UNIX to come up as a single—user system with I/O on
this device.
Note that with this first Unix release, spawning the login processes to each serial line was hardcoded and done by the init process itself. Selecting which mode to use was done before booting and switching to multi-user mode was done by modifying the switch settings and exiting the single user mode shell.
Later, with Unix Version 7 (1979), instead of hardware switches, the system was booting first in single user mode and when the single user shell exited, it switched to multi-user mode.
Even later, System III (1981) introduced the inittab file. With it, it was possible to better define and configure multiple run levels and select the one to use. Run level 1 was single-user and run level 2 multi-user. If the inittab file was missing, the system booted in single-user mode.
Best Answer
Unix was originally a product, first developed in AT&T's Bell Labs. But today, the word “Unix”, except in historical context, means a family of operating systems, not a single product (similarly to “Linux” meaning a family of distributions, not a single product). This family has a somewhat complex history (see also Evolution of Operating systems from Unix).
It's difficult to say when this product ended, because the original code was licensed to a number of vendors, some of which still maintain their product. I believe the last product released by AT&T was Unix Time-Sharing System 10 in 1989. However by that time most Unix systems were actually modified versions of the AT&T code maintained by other companies such as Sun (SunOS, later renamed Solaris), Hewlett-Packard (HP-UX), IBM (AIX), etc. These three are still extant today, and they're directly derived from the AT&T code (although after over 25 years there probably isn't much of the AT&T era code remaining).
In addition to unix systems that are derived from the AT&T code, there are systems that don't contain any AT&T code but have a similar design and compatible user and programmer interfaces. The main families of such unix systems are BSD (FreeBSD, OpenBSD, NetBSD, Darwin/macOS etc.), Linux (many distributions) and MINIX.
Just to add to the confusion, UNIXⓇ is a trademark (actually, a family of trademarks: there are several versions) which does not designate a particular product: any product can use it as long as it passes a series of conformance tests. This unusual situation is a consequence of a long legal battle. Basically, a product can lay claim to one of the UNIX trademarks if it complies with the Single UNIX Specification, which describes user and programmer interfaces of the operating system (but not administration interfaces).
If you want to run a product that came directly from AT&T, you can run Unix V5, V6 or V7 on a PDP-11 simulator (the PDP-11 was a popular series of minicomputers in from the early 1970s to the early 1990s).
If you want to run a product based on code from AT&T, you can run OpenIndiana, which is based on the now-discontinued open source edition of Solaris (OpenSolaris). OpenIndiana is free software and runs on a PC. (It might not support as much hardware as Linux does though, but it can run in e.g. VirtualBox.) I believe that you can also download Oracle's Solaris for free for personal use, and it too can run on a PC. As far as I know, it isn't possible to run AIX or HP-UX on easily-available hardware or emulators.
If you want to run a product which has the UNIX brand, you can go through the official list. It includes several versions of Solaris (including PC versions), several versions of macOS, and a few uncommon Linux distributions. In a twist of fate none of the historical Unix products have the UNIX trademark, because they're too old and fail to meet some of the newer requirements for the UNIX brand.
If you want to run a product in the unix family of operating systems, Linux is one (or rather Linux is a subfamily, and each distribution is a Unix-like operating system).