In the site-starts I was committed to, I was always present already in the private beta phase. But now, in the case of the http://ai.stackexchange.com, I couldn't.
Can my commitment still be fulfilled?
Yes you can absolutely fulfill commitment well after Private Beta. You have 180 days (6 months) to fulfill commitment, and Private Beta lasts for only 2-3 weeks. Remember also that when you clicked commit, it said "I agree to post at least 10 questions or answers within the first 6 months of the Beta", and did not say it has to be "private" beta.
This answer: https://area51.meta.stackexchange.com/a/4241/190792 says that you can even fulfill commitment again when the site leaves public beta (i.e. "graduates"), but I have not seen this happen myself, because most people that fullfill commitment, do it during the first 6 months of Beta, and not several years later when the site graduates.