In the Fission chat, you mention that people are concerned with privacy. Here you’ve suggested “privacy app”. I think this has connotations that don’t match. What about a “private app”?
I think that ideally the progression of “unhosted” apps will necessarily end up with the app itself being included in the user’s data—just like most software from the 90s was supposed to be installed on the user’s disk and kept there. In that case, it could be called a truly private app—there’s really no need to even hit the app developer’s public endpoint to use it, nor is there any problem if they’re unable to keep their site up in the long term—because your storage provider has your copy, anyway.
The notion of privacy is something that’s getting a lot of attention right now. It’s too attractive to pass up the opportunity to adopt messaging based on it, I think.
(NB: I regret that my first comment here is a form of bikeshedding, but there’s a real need for a term that’s approachable.)