The private story is an additional layer of security on Snapchat.
Snapchat represents the private story lock with the purple color. So, whenever you see a purple color lock on someone's story, you know that it's a private story.
No, you can not add a personal lock on someone's chat on Snapchat. Although you can lock your Snapchat app using third-party apps.
The third-party apps can be available on the google play store or the Apple store for iPhones.
When you see someone's private story from the story page then it will have a purple color lock
But when you will see a new private story on the chat list then you will see a blue-colored lock behind their name.
So, it's just the different colors for different pages. It does not mean anything different. It is still a private story.
You can find the stories from the chat list as well as from the story page. On the story page, you can see the stories of all your friends all together represented by a circle.
All the stories including the private stories are present on the same page.
Yes, muting someone's story means that you don't want to see that person's story. So, his stories want to appear with the other stories but you can still choose to see them.
A private story is something when you restrict the person from seeing your story.