I just noticed that there are now three different modes in Stories section of Duolingo.
They changed Listen into Conversation, but Listen is still available if you change the end of the address from ?mode=conversation to ?mode=listen.
So the types are:
Read = whole story is readable. All questions can be answered by clicking correct answers.
Listen = Some parts of the story need to be listened (can be read by clicking reveal). All questions can be answered by clicking correct answers.
Conversation = Some parts of the story need to be listened (can be read by clicking reveal). Some questions can be answered by speaking (can be changed to normal clickable ones by clicking that you don't have microphone).
I wonder when they will add typable answers.