When Snapchat records a story view
Inside the Snapchat app, viewing a story while signed in can add your account to the owner's viewer list. That is normal Snapchat behavior.
Why public web viewing is different
Public profiles can expose story media outside the app. If you watch that public media from a browser without signing in, Snapchat does not have your account identity to attach to the view.
Private stories still behave differently
A friend-only story is not public web content. To watch it in Snapchat, you need an account with access, and that account may appear in the viewer list.
For screenshots and downloads, read the Snapchat screenshot notification guide. To test a public profile, use the story viewer.