Yes indeed. One sentence (the Duolingo one) seems to indicate that the decision to remain here (or not) has not been made (yet). The other (Beppe's above) -- "Ho deciso di non rimanere qui" -- indicates that the decision HAS been made and it is to not remain here. Two quite different meanings...
As far as the English meaning: I think "I have not decided to remain here." and "I didn't decide to remain here." both mean the exact same thing. They both mean (or imply) that either the decision has not been made yet whether to stay here or not, or some other decision has been made which is not to remain here. You could follow up either sentence with "I have also not decided to leave this place." and it would not be a contradiction. It just seems like an indecisive person to me.
If you were to say, "I decided to not remain here" that would have a different meaning.
The meanings are different. "I didn't decide to remain here" would refer to some event in the past. "I didn't decide to remain here, my Mum made me stay" "I have not decided to remain here" refers more to your current decision state. Sometimes people say it when they mean "I have decided not to stay here", although the senses are technically different. "Are you keeping your room?" "I have not decided to remain here" means don't give me a contract yet. " I have decided not to stay here is unambiguous
Thank you - while we can discuss the different interpretations, it's most helpful to have someone clarify how to say the second sentence in Italian. Putting non in front of rimanere makes sense in where the emphasis is. Have a lingot for getting to the heart of the matter!