I disagree. I believe that it’s more common for “private” to mean “between you and the company”. Eg. Private accounts on other services, or private information associated with your account.
Plenty of people had “private” LiveJournal accounts back in the day with no expectation that it was a zero-knowledge system. Just that they weren’t sharing it. I may not be a good indication of “most people”, but I just don’t think that most people think of privacy in the way that we (folks who know details of encryption and metadata leaks) do.
But that's the problem, isn't it? I mean, many people do (sort of magically) think that their online stuff is private. And then they get pwned in one way or another.
As I read that, users must trust you. There's nothing that would actually prevent you from accessing the data.
So it's arguably misleading to call it a "private diary".