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Not all stories have this icon and I can’t figure it out.
The icon is for lists. The list function allows you to tag a story that you haven't read yet and save it to read later. It's different from favorites, which are stories you've read and particularly like.
That distinction has been here for a while now. I don't think there's a function to automatically remove a story from a list once it's been read, so there's a subtle distinction between marking a story as "favorite" and bookmarking it.