When you are watching a lecture video and turn on subtitles, a localstorage setting is saved as kk-vimeo-player. This contains the language setting as an object. If you change the language, say, from English to French, the setting is applied and saved there. However, if you turn off the subtitles (clicking CC → “Off”), this language object will not be deleted. When you go watch the next video, the subtitles will appear again. To fix this issue, I think the language object within kk-vimeo-player should be removed when clicking “Off” in the subtitle language settings in order to actually turn off subtitles when no longer needed.
I see this less as “an issue” but more as an implementation choices. Most people who need subtitles will need them consistently, so for those users, the behavior sounds fairly optimal – you may or may not use subtitles all the time, but in general, it will be in one language, say, as in this case, French.
You have the disadvantage of having the less used “use case” 
Got it. So, following from this implementation choice, there is no way to turn off the subtitles except manually manipulating the browser’s local storage. Fair enough. I just wanted to point this out as the behaviour is quite different compared to other video platforms, where you can switch languages or turn off subtitles persistently. Thanks for the reply!