Exclusive Snap Diary. A diary is something personal and private, and by using 'exclusive' it emphasizes that only you or a select few are meant to see the snap story. It gives a sense of intimacy.
The Private Snap Archive. This name suggests that your snap story is like an archive, a place where you store and preserve your private snaps, as if they were important historical records or memories.
Personal Snap Vault
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.
How about 'My Mysterious Musings', 'Sneaky Snaps' or 'Covert Craziness'?
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.
'Secret Chronicles', 'Hidden Tales', 'Personal Reveries' are great creative names for private stories.
How about 'Secret Silliness', 'Hidden Hilarity' or 'Private Pranks'?
Think about your personality. If you're a bit of a dreamer, names like 'My Dreamy Diaries' could work. If you're more of an adventurer, 'Adventure Archives' might be good.
You could use names like 'My Secret Moments', 'Personal Chronicles', or 'Exclusive Tales'.