Mentioned somewhere else: Windows NT has subsystems. E.g. posix, os2, win32, ... What they do is adding a subsystem to windows. The module they code is the subsystem not the distribution. Nowadays it is a bit blurry die to the full kernel and this windowing subsystem.

AFAIK it was their legal department, who advised them that calling it a "Linux Subsystem" may run afoul of the Linux trademark, especially considering that in WSL1, Linux itself was not involved at all.

Correct. It's a trademark law issue. In the same way that you can develop a "Chess for Windows" but not a "Windows Chess". Using "for X" is fair use.

You say "correct" but you're directly contradicting the parent's point, since with your argument the easily and correctly parsed 'Linux Subsystem "for Windows"' would not run afoul of any trademarks.

Remember -- the name "Linux" is trademarked. Microsoft is free to call something Windows X since they own the Windows trademark, but they can't call something Linux X because they don't own the Linux trademark. But they can call something X for Linux the same way a third party can call something X for Windows. Hence, Windows Subsystem for Linux. It sounds dumb, but that's the legal rationale.

Similarly, the unofficial Reddit apps had to rename from e.g. "Reddit Sync" to "Sync for Reddit".

But even in your example, it shows how bad of a name WSL is. "Chess for Windows" implies that there is a chess program that runs on windows. By analogy, "Windows Subsystem for Linux" implies that there is a Windows Subsystem that can run on Linux.