Personal Chronicles. This name gives the idea that your private snap story is like a personal diary, chronicling your unique experiences, thoughts, and feelings that you only want to share with a select few. It has a bit of a nostalgic and personal touch to it.
Some other great names might be 'Exclusive Exposures', 'Hidden Highlights', or 'Confidential Chronicles'. These names give a sense of privacy and exclusivity which is perfect for a private story. The idea is to choose a name that reflects the personal nature of the content that will be shared in the snap private story.
Forbidden Fragments. The word 'forbidden' gives an air of secrecy, and 'fragments' can imply that the snap story is made up of different parts of your private life that are not meant for public consumption. This name makes it sound alluring and private at the same time.
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.