You can also consider names like 'Exclusive Insights', which gives the idea that it's something only a select few can see. Another option is 'Private Perspectives', as it implies that it's your own view of things and is private. 'Confidential Chronicles' is also a great name as it has a sense of secrecy and a story - like quality.
Private Moments Gallery. This name gives the idea that your snap story is a collection of personal and private moments that are precious to you. It has a bit of an artistic and exclusive feel to it.
A good name for a snap private story should be concise. Long and complicated names are hard to remember. It should also be relevant to your personality or the theme of the stories. For instance, if you are a very creative person and your private stories are about your art projects, 'Creative Corner - Private' would be a great name. Additionally, it can be something that makes you feel a connection when you see it, like a name that has a special meaning only to you or your close friends.