Is it true that Tahajjud prayer is the real Taraveeh prayer?
Yes, Tahajjud prayer is the real Taraveeh prayer.
Taraveeh prayer is the special prayer ordained for the month of Ramadan. It has to be performed each night during the month of Ramadan. It is in fact offered at Tahajjud time.
The observance of Taraveeh prayer after the Isha prayers was allowed during the Khilafat of Hazrat Umar(ra), to enable such people, who for unavoidable reasons could not perform Taraveeh prayer at Tahajjud time (in pre-dawn hours), to still offer this prayer. However, it is preferable to offer this prayer at Tahajjud time.
A hadith to support this interpretation can be found in Muslim. It is narrated by Abu Huraira(ra) that:
‘Allah’s Apostle(saw) said, “Whoever prayed at night the whole month of Ramadan out of sincere faith and hoping for a reward from Allah, then all his previous sins will be forgiven.”’ (Muslim)
Abdur Rahman bin Abdul Qari said,
‘I went out in the company of Umar bin Al-Khattab one night in Ramadan to the mosque and found the people praying in different groups – a man praying alone or a man praying with a little group behind him. So, Umar said, “In my opinion I would better collect these (people) under the leadership of one Qari (Reciter) (i.e. let them pray in congregation!)”. So, he made up his mind to congregate them behind Ubai bin Ka’b. Then on another night I went again in his company and the people were praying behind their Qari. On that, Umar remarked, “What an excellent Bid’a (i.e. innovation in religion) this is; but the prayer which they do not perform, but sleep at its time is better than the one they are offering.” He meant the Tahajjud prayer in the last part of the night. In those days people used to pray in the early part of the night.’ (Bukhari)
It is important to note that if Taraveeh prayer is offered after Isha prayer then Muslims should still try to offer Tahajjud prayer as well, as that is a permanent Sunnah of the Holy Prophet(sa) for whole year.
Generally the Holy Prophet(sa) used to offer eight rakats for Tahajjud prayer (so the same applies to Taraveeh prayer) but if for some reason fewer rakats are offered (in sets of two rakats) then this is also permissible.