How exactly did Windows become the OS of the home PC?
Is it true that the Windows OS, at it's core was originally designed to simply be the OS of the terminals of the windows server architecture.
No. Windows started out as a GUI component of DOS - eventually hiding DOS beneath. Anything like a windows server architecture was only devloped way later. In fact, much later than the time Windows became a stand alone OS (wich is, depending on your PoV OS/2 or Windows NT)
So if this is true how exactly did a OS intended to support a companies server offerings become the defacto standard of home computing?
Because it is not true. Windows became a thing for home users way before any server architecture and mostly thru several factors:
It offered a GUI to existing PC users. At that point it was in fact rivalled by GEM, which was, for some time quite ahead in sales of Windows, at least in Europe that is, or more integrated solutions like DeskMate or late comer GEOS.
PC-Hardware became rather cheap during the late 1980s, early 1980s1990s, eventually undercutting equal powerful high end home computers like Atari ST, Amiga or Acorn. Windows was the default counterpart to their offerings for a GUI
During the mid to late 1990 the PC became a serious game platform, even more rivalling existing home computer platforms. With evolving graphics and sound capabilities, as well as (at the time) almost endless memory, 486 machines draw in game development - not to mention the ability of DOS for CD support, easing the distributing of ever growing games. This again cemented the PC as base hardware and while many of these games were DOS based using various extenders (DOS/4GW made this race due being included with Watcom C), Windows was a simple choice for all other tasks - even more so when game companies offered kind of integration to the desktop.
The most important, single turning point by Microsoft was the introduction of DirectX in 1995/96, with Windows 95. While it wasn't complete new (there was WinG before), it was the dedication to a unified set of APIs to cover all (well, most) hardware out there - this came at the same time when the 32 Bit API was stabilized, making DOS-Extenders obsolete, creating the perfect wave for game developers to switch - which in turn made even hard core DOS users install Windows - and the remaining home computer users switch.
Last, but for sure not least, the DOS/Windows-Tax, that forced system manufacturers to either supply some alternate OS (and still pay to some degree) or simply deliver their machines with DOS/Windows. This was heavily enforced by MS during the late 1990s, thus making next to every brand machine being delivered with Windows installed by default. Users had to active change this - something most never even think of.
So in conclusion, Windows 3.1 became a usable in ~1993 as (application) Desktop, Windows 95 made 1996 the turning year, were DirectX was able to pull developers over from DOS. And it proved to be a cornerstone up till today. All with a more or less gentle nudging by lawyers waving their version of a 2x4.