Private Chronicles. Chronicles usually refer to a series of events or stories. By adding 'private', it makes it clear that these are personal and not for general viewing.
My Secret Moments. It gives the idea that the story contains personal and private happenings that are only for a select few to see.
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 could go for something like 'My Secret World' or 'Hidden Moments'. They give a sense of privacy and mystery.
'My Secret Moments', it's simple and directly shows that it's about your private moments.
Well, how about 'Behind the Scenes' or 'Personal Glimpses'? They give a sense of privacy and exclusivity.
Well, 'Behind the Scenes' is a nice option. It gives the idea that there are things not usually seen. Another one could be 'Personal Paradise', suggesting a special and private place for you.