B – As for his actually leading the prayer himself, there are several ḥadiths:

The first

• 2 (ṣaḥīḥ)
  Nuʿmān ibn Bashīr reported: “We stood in prayer with the Messenger of Allah ﷺ on the night of the twenty-third of Ramadan until one third of the first part of the night had elapsed. We then stood with him on the night of the twenty-fifth until half the night had passed, and on the night of the twenty-seventh he led us until we thought we would miss al-falāḥ—that was what we used to call the pre-dawn meal of suḥūr.”

Narrated by Ibn Abī Shaybah in the Muṣannaf (2 / 90 / 2), Ibn Naṣr (p. 89), al-Nasaʾī (1 / 238), Aḥmad (4 / 272), and al-Firyānī. Its isnād is sound, and al-Ḥākim (1 / 440) authenticated it, saying:
“In it is a clear proof that offering Tarawih in the mosques of the Muslims is an established sunnah. ʿAlī ibn Abī Ṭālib used to urge ʿUmar, may Allah be pleased with them both, to revive this sunnah until he finally did so.”

The second

• 3 (ṣaḥīḥ)
  Anas reported: “The Messenger of Allah ﷺ was praying in Ramadan. I came and stood beside him; then another man came, and another, until we were a small group ⦗1⦘. When the Messenger of Allah ﷺ realized that we were behind him, he shortened ⦗2⦘ the prayer and then entered his house. Once inside, he performed a prayer he had not performed with us. When morning came we said, ‘Messenger of Allah, did you notice us last night?’ He replied, ‘Yes, and that is what made me do what I did.’”

Narrated by Aḥmad and Ibn Naṣr through two sound chains, and by al-Ṭabarānī in al-Awsat with a similar wording.