I am the most knowledgeable of people about the time of this prayer. The Messenger of Allah ﷺ used to perform it when the moon set on the third night.
حدثنا محمد بن عبد الملك بن أبي الشوارب حدثنا أبو عوانة عن أبي بشر عن بشير بن ثابت عن حبيب بن سالم
أنا أعلم الناس بوقت هذه الصلاة كان رسول الله ﷺ يصليها لسقوط القمر لثالثة
[Its isnad is sahih] Abu Dawud also reported it in the Book of Prayer, chapter on the time of the last night prayer, no. 419, from the hadith of Abu Awana with the same wording. Ibn Hibban graded it sahih (al-Ihsan), no. 1524, as did Hakim, 1/194, and Dhahabi agreed with him.