Is it true that Salatul-Tahajjud is an innovation (bid‘ah) and that there is no distinction between it and Tarawih, and that the first to make this distinction was the Imam of the Haram, Abdullah Al-Khulaifi, about 50 years ago, and what is the ruling on praying it separately?

Night vigil prayer (Qiyam al-Layl) is recommended both inside and outside of Ramadan, though it is more emphasized during Ramadan, and congregational prayer is legislated for it. Its time extends from after the 'Isha prayer until the appearance of dawn, and there is no specific number of rak'ahs for it. The prayer of Qiyam is called Tarawih, and Tahajjud is Qiyam, though it is also said to be Qiyam performed after sleep. The entire night is a time for Qiyam, and there is no objection to dividing Qiyam into two parts for ease and to increase good deeds. This is what Muslims have practiced throughout history, and it is the view of the majority of scholars, contrary to those who forbade increasing beyond eleven rak'ahs. As long as the entire night is a time for Qiyam, and separation makes it easier, there is no objection to dividing Qiyam into two parts.