Apple TV is not Apple TV+?

I own movies in my Apple account and can play them on iOS with Apple TV. Yet when I log into Apple TV on my cable box, I am not allowed to access that library. I can only watch their streaming programs.

That is how this apparently was negotiated between Apple and the cable company. A specific app for this one streaming service, but not the additional Apple media services that might cut into what the cable company sells.

So in what sense do I own these movies? Only in certain versions of Apple TV?

Your purchased movies are yours in the same way as when you purchased them. Available to watch on Apple devices with the (full) Apple TV app or iTunes application, and also on select third party devices with the (full) Apple TV app.

In which case how can they call the cable box version of their app “Apple TV+” when it’s actually minus an entire feature of viewing purchased films?

The Plus in the app name is about the streaming services named Apple TV+. The app name ‘Apple TV+ app’ reflects that is it for that service specifically, not including the other Apple media services. Get the Apple TV app or Apple TV+ app on your smart TV or streaming device - Apple Support. This support document has more on the differences.

no Apple TV app on Windows,

Currently not. Some new Apple apps for Windows have been announced for 2023, though.

iTunes app, which doesn’t launch my movies

Sorry to hear that. It does work for many other users, so apparently something is off about your system, settings, or setup. Try the iTunes for Windows forum for more expertise.