Is haven't supposed to mean something I may still do and didn't not?
Can I use didn't for something I might still do?
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Haven't and didn't are different time-wise, as you have guessed correctly. Haven't refers to the past up until now. So if you haven't done something, you haven't done it for a specific period of time (day, month, ever, etc.)
Didn't refers to a specific point of time that has already passed. For example, if it is 7 PM, you could say "I didn't eat dinner at 6" or "I haven't eaten dinner yet today". In both instances, you can still eat dinner. Plus, it is possible for you to have eaten dinner at a different time if you use didn't, but not with haven't, since you specified today.
When answering a question, take your cue from the form used in the question and answer accordingly. For example, "Have you been out today?" expects an answer like "No, I haven't (yet)" as opposed to "No I didn't", which would sound wrong.
"Did you write that letter?" expects a response such as "Yes, I did", and not "Yes, I have."