Is it obligatory for me to offer a Hady (sacrificial animal) because I did not stay overnight in Muzdalifah, after I had prayed Maghrib and Isha there, combined and shortened, thinking it was Muzdalifah, then discovered later that the place was not Muzdalifah?

Staying overnight in Muzdalifah is one of the obligatory rites of Hajj. A compensatory sacrifice (dam) is required from anyone who misses it, whether intentionally or unintentionally, knowingly or unknowingly, because they have abandoned a ritual. Whoever stands in a place they thought was Muzdalifah, then it becomes clear that their assumption was incorrect, is like someone who did not stand there at all. Therefore, whoever is not certain of having stood in Muzdalifah must sacrifice a sheep, to be distributed among the poor of the Haram (definition), for each individual.