I use both and I haven’t found it too confusing, to be honest. I just think of it as Apple TV (streaming device) gives access to Apple TV+ (streaming service).
Hangouts, Allo, Duo, Buzz, Google Talk, GChat, Inbox, Messenger, Messages, Bard, Gemini, etc. Who else has a track record of chopping and changing like this?
To you. You are projecting.
Apple TV (app) gives access to Apple TV+ (streaming service)