Earlier quoted context omitted.
That one isn't an example of this. It is actually a Windows Subsystem (at least WSL1) that exposes Linux syscalls, so is for Linux userspace programs. There is also the Windows Subsystem for Win32 and there used to be a Windows Subsystem for Unix. Linux Subsystem would be completely wrong, because it is a Subsystem of Windows not of Linux.
No it wouldn't. Following the scheme a couple of comments above, we have: Y of X providing Z - Windows Subsystem for Linux. Y providing X on Z - Linux Subsystem for Windows. The former is "for [having]", the latter is "for [use on]".
MS has some confusing naming, this isn't one of them.