Agree that language is really helpful here, and I also think the SAFE App in itself is a huge step towards this problem.
If a user can easily access their files through the SAFE App as well as through the app with which they created them, then that in itself shows they own the files, (as opposed to the 3rd party app,) much in the same way that if a photo is in the photos folder in my OS, then I know by default that I own it, rather than Instagram or whoever. I think the idea of the SAFE App being like an online Operating System is really powerful in subconsciously communicating these ideas.